Published by Rankguru Technology Solutions Private Limited
IL Ranker Series Mathematics for JEE Grade 11 Module 5
ISBN 978-81-983575-6-4 [SECOND
EDITION]
This book is intended for educational purposes only. The information contained herein is provided on an “as-is” and “as-available” basis without any representations or warranties, express or implied. The authors (including any affiliated organisations) and publishers make no representations or warranties in relation to the accuracy, completeness, or suitability of the information contained in this book for any purpose.
The authors (including any affiliated organisations) and publishers of the book have made reasonable efforts to ensure the accuracy and completeness of the content and information contained in this book. However, the authors (including any affiliated organisations) and publishers make no warranties or representations regarding the accuracy, completeness, or suitability for any purpose of the information contained in this book, including without limitation, any implied warranties of merchantability and fitness for a particular purpose, and non-infringement. The authors (including any affiliated organisations) and publishers disclaim any liability or responsibility for any errors, omissions, or inaccuracies in the content or information provided in this book.
This book does not constitute legal, professional, or academic advice, and readers are encouraged to seek appropriate professional and academic advice before making any decisions based on the information contained in this book. The authors (including any affiliated organisations) and publishers disclaim any liability or responsibility for any decisions made based on the information provided in this book.
The authors (including any affiliated organisations) and publishers disclaim any and all liability, loss, or risk incurred as a consequence, directly or indirectly, of the use and/or application of any of the contents or information contained in this book. The inclusion of any references or links to external sources does not imply endorsement or validation by the authors (including any affiliated organisations) and publishers of the same.
All trademarks, service marks, trade names, and product names mentioned in this book are the property of their respective owners and are used for identification purposes only.
No part of this publication may be reproduced, stored, or transmitted in any form or by any means, including without limitation, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the authors (including any affiliated organisations) and publishers.
The authors (including any affiliated organisations) and publishers shall make commercially reasonable efforts to rectify any errors or omissions in the future editions of the book that may be brought to their notice from time to time.
All disputes subject to Hyderabad jurisdiction only.
Dr. B. S. Rao, the visionary behind Sri Chaitanya Educational Institutions, is widely recognised for his significant contributions to education. His focus on providing high-quality education, especially in preparing students for JEE and NEET entrance exams, has positively impacted numerous lives. The creation of the IL Ranker Series is inspired by Dr. Rao’s vision. It aims to assist aspirants in realising their ambitions.
Dr. Rao’s influence transcends physical institutions; his efforts have sparked intellectual curiosity, highlighting that education is a journey of empowerment and pursuit of excellence. His adoption of modern teaching techniques and technology has empowered students, breaking through traditional educational constraints.
As we pay homage to Dr. B. S. Rao’s enduring legacy, we acknowledge the privilege of contributing to the continuation of his vision. His remarkable journey serves as a poignant reminder of the profound impact education can have on individuals and societies.
With gratitude and inspiration
Team Infinity Learn by Sri Chaitanya
Key Features of the Book
Chapter Outline
1.1 Introduction to Relations
1.2 Types of Relations
1.3 Introduction to Functions
1.4 Kinds of Functions
1.5 Composition of a Function
1.6 Inverse of a Function
This outlines topics or learning outcomes students can gain from studying the chapter. It sets a framework for study and a roadmap for learning.
Specific problems are presented along with their solutions, explaining the application of principles covered in the textbook. Solved Examples
Try yourself:
1. If () 4 nA = and () 3 nB = , then find the number of onto functions from the set A to the set B .
Ans: 36
1. Is ) :0,fR →∞ defined by () 2 fxx = onto?
Sol: Yes, the function is onto because for every positive real number y , there exists a real number x such that () fxy = , where 2 yx = .
Try Yourself enables the student to practice the concept learned immediately.
This comprehensive set of questions enables students to assess their learning. It helps them to identify areas for improvement and consolidate their mastery of the topic through active recall and practical application.
CHAPTER REVIEW
Introduction to Relations
1. R is said to be a relation on the set A if ⊆× RAA
TEST YOURSELF
1. If the system of equations 2x – 3y + 4 = 0, 5 x –2 y –1=0 and 21 x – 8 y + λ = 0 are consistent, then λ is (1) –1 (2) 0 (3) 1 (4) –5
It offers a concise overview of the chapter’s key points, acting as a quick revision tool before tests.
Organised as per the topics covered in the chapter and divided into three levels, this series of questions enables rigorous practice and application of learning.
These questions deepen the understanding of concepts and strengthen the interpretation of theoretical learning. These complex questions combining fun and critical thinking are aimed at fostering higher-order thinking skills and encouraging analytical reasoning.
Exercises
JEE MAIN LEVEL
LEVEL 1, 2, and 3
Single Option Correct MCQs
Numerical Value Questions
THEORY-BASED QUESTIONS
Very Short Answer Questions
Statement Type Questions
Assertion and Reason Question
JEE ADVANCED LEVEL
BRAIN TEASERS
FLASHBACK
CHAPTER TEST
This comprehensive test is modelled after the JEE examination format to evaluate students’ proficiency across all topics covered, replicating the structure and rigour of the JEE examination. By taking this chapter test, students undergo a final evaluation, identifying their strengths and areas of improvement.
Level 1 questions test the fundamentals and help fortify the basics of concepts. Level 2 questions are higher in complexity and require deeper understanding of concepts. Level 3 questions perk up the rigour further with more complex and multi-concept questions.
This section contains special question types that focus on in-depth knowledge of concepts, analytical reasoning, and problem-solving skills needed to succeed in JEE Advanced.
Handpicked previous JEE questions familiarise students with the various question types, styles, and recent trends in JEE examinations, enhancing students’ overall preparedness for JEE.
CIRCLES CHAPTER 13
Chapter Outline
13.1 Equation of Circle
13.2 Tangent and Normal
13.3 Pair of Tangents
13.4 Relative Position of Two Circles
13.5 System of Circles
The study of equation of circles focuses on algebraic representations of circles. We learn to derive equations using the centre-radius form and understand key elements such as the centre coordinates and radius. This knowledge is essential for solving geometric problems and analysing circle-related phenomena in mathematics and physics.
13.1 EQUATION OF CIRCLE
In a plane, the locus of a point which moves at a constant distance from a fixed point is called a circle. The fixed point is called centre C of the circle, and the constant distance is called radius r of the circle.
1. If r = 0, then the circle is called point circle.
2. If r > 0, then the circle is called real circle.
3. If r < 0, then the circle is called imaginary circle.
A circle of radius one unit is called unit circle and a circle of radius zero is called point circle.
13.1.1 Standard Form of Circle
Equation of circle having centre at ( a, b) and radius r is (x – a)2 + (y – b)2 = r2
Equation of circle having centre at origin and radius r is x2 + y2 = r2; this is called standard form of equation of the circle.
Equation of circle having centre at origin and radius 1 unit is x 2 + y 2 = 1; it is called unit circle.
The second-degree equation in terms of x, y is ax2 + 2hxy + by2 + 2gx + 2fy + c = 0; the condition that this equation represents a circle are
i. a = b ≠ 0
ii. h = 0
iii. g2 + f2 – ac ≥ 0
1. Find the equation of a circle having centre at (2, 3) and radius 13 units.
Sol: Substitute a = 2, b = 3, r = 13 in the equation (x – a)2 + (y – b)2 = r2
Hence, the equation of circle is (x – 2)2 + (y – 3)2 = 132
Simplifying,
x2 + y2 – 4x – 6y – 156 = 0
Try yourself:
1. Find the equation of a circle having centre at orgin and radius 5 units.
Ans: x2 + y2 = 5
13.1.2 General Equation of the Circle
1. The general equation of the circle is S ≡ x2 + y2 + 2gx + 2fy + c = 0; g2 + f2 – c ≥ 0
2. The centre and radius of the circle x 2 + y 2 + 2 gx + 2 fy + c = 0 are C (– g , – f ), 22 rgfc =+−
If g2 +f2 – c = 0, then the equation x2 + y2 + 2gx + 2fy + c = 0 represents a point circle and that point is (– g, –f).
3. The equation of circle passing through the origin is in the form of x2 + y2 + 2gx + 2fy = 0.
4. The equation of circle having centre on x-axis is x2 + y2 + 2gx + c = 0.
5. The equation of circle having centre on y-axis is x2 + y2 + 2fy + c = 0.
6. For the circle ax2 + ay2 + 2gx + 2fy+c =
0, the centre is , gf aa and radius is 22 gfac r a +− =
2. Find the centre and radius of the circle x2 + y2 + 6x – 4y + 4 = 0.
Sol: Comparing the equation with general form of circle, we get g = 3, f = –2, c = 4
Hence, centre is C(–3,2) and radius is ()2 2 3243 r =+−−=
Try yourself:
2. Find the equation of a circle with centre at origin and passing through the point of intersection of the lines 3 x – 2y – 1 = 0, 4x + y – 27 = 0.
Ans: x2 + y2 = 74
13.1.3 Concentric Circles
Two circles are said to be concentric if and only if they have the same centre and different radii. C r1 r2
1. Concentric circles will not intersect each other.
2. Equations of concentric circles differ by a constant.
3. The equation of the circle concentric with x2 + y2 + 2gx + 2fy + c = 0 is x2 + y2 + 2gx + 2fy + k = 0, where k is constant such that g2 + f2 – k ≥ 0.
Concentric circles are coincident if their radii are equal.
3. Find the equation of circle concentric with x2 + y2 –3x + 4y – c = 0 and passing through the point (–1, –2).
Sol: The equation of required circle is x2 + y2 –3x + 4y+k= 0 Since it passes through the point (–1 ,–2),
⇒ 1 + 4 – 3(–1) + 4(–2) + k = 0
⇒ k = 0
Therefore, required circle equation is x2 + y2 –3x + 4y= 0.
Try yourself:
3. Find the equation of a circle having area 22 square units and concentric with x2 + y2 + 8x + 12y + 15 = 0.
Ans: x2 + y2 + 8x + 12y + 45 =0
Notations:
■ S ≡ x2 + y2 + 2gx + 2fy + c
■ S1 ≡ xx1 + yy1 + g(x + x1) + f(y + y1) + c
■ S11 ≡ x1 2 + y1 2 + 2gx1 + 2fy1 + c
■ S12 ≡ x1x2 +
+ c
13.1.4 Chord
Let P and Q be any two points on a circle, as shown below.
Sol: Equilateral triangle inscribed in the circle is as shown below. (0, 0) 2a a O D A
Chord PQ
1. The line that passes through P and Q is called secant
2. The line segment PQ is called chord of the circle
3. Foot of the perpendicular from centre of the circle to the chord PQ is midpoint of PQ
4. Angle made by any chord at the centre is double the angle made by that chord at any point on the same side of the arc.
5. If the chord PQ is passing through the centre of the circle, then PQ is called diameter of the circle, this is largest chord of the circle.
6. The locus of midpoints of parallel chords of a circle is called diameter of that circle.
7. Equation of chord joining two points
P(x1, y1) and Q(x2, y2) on the circle S = 0 is S1 + S2 = S12.
8. The length of the chord L = 0 of a circle having centre at C and radius r is 22 2 rd , where d is perpendicular distance from centre C to the line L = 0.
= 0
4. Find the equation of a circle having centre at origin and circumscribing an equilateral triangle whose length of the median is 3 a.
Given: The centre of the circle is origin, and the length of the median is 3a.
In equilateral triangle, circumcentre is same as centroid, O divides AD in the ratio 2:1, internally.
Hence, r = 2a
Therefore, the equation of circle is x2 + y2 = 4a2 .
5 Find the length of the chord intercepted by the circle x2 + y2 –x + 3y – 22 = 0 on the line y = x –3.
Sol: The given circle is x2 + y2 –x + 3y – 22 = 0. Centre is 13 , 22
2 r =++=
The perpendicular distance from the centre 13 , 22
of the circle to the line y = x –3 is
Therefore, the length of the chord is
Try yourself:
4. Two parallel chords are drawn on the same side of the centre of a circle of radius 12 units. It is found that they subtend 72° and 144° at the centre of the circle. Find the perpendicular distance between the two chords.
Ans: 6
13.1.5 Equation of a Circle when Diameter
End Points Are Given
Equation of a circle having P( x 1 , y 1 ) and Q(x2, y2) as ends of diameter is (x – x1)(x – x2) + (y – y1)(y – y2) = 0.
1. Centre of the circle is midpoint of PQ.
2. Radius is 2 PQ
Every diameter of the circle cuts the circle into two equal parts. Each part is called semicircle, and angle in semicircle is 90°.
If x 1 , x 2 are the roots of the equation x2 +ax + b = 0 and y1, y2 are the roots of equation y2 +cy + d = 0, then equation of a circle having line joining two points A(x1, y1), B(x2, y2) as diameter is x2 + y2 + ax + cy + b + d = 0.
There are infinite number of circles passing through the given two points. Among all these circles, the smallest circle is a circle having these two points as ends of diameter.
Equation of any circle passing through the points of intersection of a circle S = 0 and a line L = 0 is S + λL = 0 , where λ is parameter.
S = 0
S + λL = 0
L = 0
6. Find the equation of the circle passing through (1, 0), (0, 1) and having smallest possible radius.
Sol: Circle passing through two points and having smallest possible radius is a circle having those two points as ends of diameter.
Equation of circle having (1, 0) and (0, 1) as ends of diameter is ( x – x 1)( x – x 2) + ( y – y 1) (y – y2) = 0.
Hence, the equation of circle is x(x – 1) + y(y – 1) = 0.
Therefore, the equation of circle is
x2 + y2 – x – y= 0.
Try yourself:
5. The equations of perpendicular bisectors of two sides AB and AC of a triangle ABC are x + y + 1 = 0 and x – y + 1 = 0, respectively. If the circumradius of triangle ABC is 2 units and the locus of the vertex A is x2 + y2 + 2x – k= 0, then find the value of k. Ans: k = 3
13.1.6 Equation of Circle Passing through Three Non-collinear Points
To find the equation of a circle passing through three points A(x1, y1), B(x2, y2) and C(x3, y3), follow the given below steps.
1. Substitute three points in the standard form of equation of circle x2 + y2 + 2gx + 2fy + c = 0 to get three equations in terms of g, f, c.
2. Solve for g, f, c.
3. Substitute g, f, c values in the standard form to get the equation of required circle.
Equation of a circle passing through the origin and (a, 0), (0, b) is x2 + y2 – ax – by = 0.
(0, b)
(a, 0)
(0, 0)
This is the circle circumscribing the triangle formed by the line 1 xy ab += with coordinate axes.
Equation of circle circumscribing the triangle formed by the line ax + by + c = 0 with coordinate axes is ab(x2 + y2) – c(bx + ay) = 0.
7 Find the equation of the circle that passes through (6, 5) and (4, 1), having diameter along the line 4x + y = 16.
Sol: Suppose, the equation of the required circle is x2 + y2 + 2gx + 2fy + c = 0.
It is passing through (6, 5), (4, 1).
Hence, 12g + 10 + c = –61 …… (1)
8g + 2f + c = –17 …… (2)
Since the centre of the circle lies on the line 4x + y = 16,
so 4g + f + 16 = 0 …… (3)
Solving the above three equations, g = 3, f = –4, c = 15
Therefore, equation of required circle is x2 + y2 –
6x – 8y + 15 = 0.
Try yourself:
6. Find the equation of the circle circumscribing the triangle formed by the lines x + y = 6, 2x + y = 4, x + 2y = 5.
Ans: x2 + y2 –17x – 19y +50 = 0
13.1.7 Concyclic Points
If four or more points lie on the same circle, then those points are called concyclic points.If the lines a 1 x + b 1 y + c 1 = 0 and a 2x + b 2y + c 2 = 0 intersect the axes at four points which are concyclic points, then a1a2 = b1b2 and the equation of circle passing through those four points is ( a1x + b1y + c1) (a2x + b2y + c2) – (a1b2 + a2b1)xy = 0. A B x D y C O
If A(a1, 0), B(a2, 0), C(0, b1), D(0, b2) are concyclic points, then a1a2 = b1b2.
8 Find the value of k, if the points (2k, 3k), (1, 0), (0, 1) and (0, 0) are concyclic.
Sol: Let A= (2 k , 3 k ), B = (1, 0), C = (0, 1), D = (0, 0).
Equation of the circle passing through the points B, C, D is x2 + y2 – x – y = 0.
Since given points are concyclic, then A lies on the above circle.
⇒ 4k2 + 9k2 – 2k – 3k = 0
⇒ 13k2 – 5k = 0
⇒ k(13k – 5) =0
⇒ 5 (0) 13 kk=≠
Try yourself:
7. If the lines λx–y+1=0 and x–2y+3=0 cut the coordinate axes at concyclic points, then find the value of λ .
Ans: 2
13.1.8 Parametric Equation of a
Circle
1. Equation of circle x2 + y2 = a2 in parametric form is x = a cos q , y = a sin q, 0 ≤ q < 2π.
2. Equation of circle x2 + y2 + 2gx + 2fy + c = 0 in parametric form is x+g = r cos q , y+f = r sin q.
3. Equation of chord joining the two points q 1, q 2 on the circle x 2 + y 2 = r 2 is
xyr
4. The length of chord AB joining A( q 1), B(q2) on the circle x2 + y2 = r2 (or) (x – x1)2 + (y – y1)2 = r2 is 12 2sin 2 r θθ
9 Write the parametric equations of the circle 2x2 + 2y2 = 7.
Sol: Given: Circle is x2 + y2 = 2 7 2
Parametric equations of circle are 77 cos,sin 22xyθθ== , where 0 ≤ q < 2π
Try yourself:
8. Find the length of the chord joining the parametric points , 26 PQ ππ
on the circle x2 + y2 – 6x – 4y – 12 = 0.
Ans: 5
TEST YOURSELF
1. The equation of the circle concentric with the circle x2 – y2 – 6x + 12y +15 = 0 and of double its area is
(1) x2 + y2 – 6x + 12y – 15 = 0
(2) x2 + y2 – 6x + 12y – 30 = 0
(3) x2 + y2 – 6x + 12y – 25 = 0
(4) x2 + y2 – 6x + 12y – 20 = 0
2. The diameters of a circle are along 2x + y – 7 = 0 and x + 3y –11 =0 . Then, the equation of the circle, which also passes through (5, 7), is
(1) x2 + y2 – 4x – 6y – 16 = 0
(2) x2 + y2 – 4x – 6y – 20 = 0
(3) x2 + y2 – 4x – 6y – 12 = 0
(4) x2 + y2 + 4x + 6y + 20 = 0
3. If the centroid of an equilateral triangle is (1, 1) and one of its vertices is (–1, 2), then equation of its circumcircle is
(1) x2 + y2 – 2x – 2y – 3 = 0
(2) x2 + y2 + 2x – 2y – 3 = 0
(3) x2 + y2 – 4x – 6y + 9 = 0
(4) x2 + y2 + x – y + 5 = 0
4. If the lines x – 2y + 3 = 0, 3x + ky + 7 = 0 cut the coordinate axes in concyclic points, then k =
(1) 3 / 2 (2) 1 / 2
(3) – 3 / 2 (4) – 4
5. If the line 3x – 2y + 6 = 0 meets at x-axis and y -axis, respectively, at the points A and B, then the equation of the circle with radius AB and centre at A is
(1) x2 + y2 + 4x + 9 = 0
(2) x2 + y2 + 4x – 9 = 0
(3) x2 + y2 + 4x + 4 = 0
(4) x2 + y2 + 4x – 4 = 0
6. If the lines 2x + 3y + 1 = 0, 6x + 4y + 1 = 0 intersect the coordinate axes at four points, then the circle passing through the points is
(1) 12x2 + 12y2 + 8x + 7y + 1 = 0
(2) 6x2 + 6y2 + 3x + y = 0
(3) 12x2 + 12y2 + 8x + 7y + 3 = 0
(4) x2 + y2 + 4x – y + 3 = 0
7. Centre and radius of the circle with segment of the line x + y = 1 cut off by coordinate axes as diameter is
(1) 111 , , 22 2
(3) 111 , , 22 2
(2)
, , 22 2
(4) 111 , , 22 2
8. The abscissae of two points A and B are the roots of the equation x2 + 2ax – b2 = 0 and their ordinates are the roots of the equation y2 + 2py – q2 = 0. Then, the radius of the circle with AB as diameter is
(1) 2222 abpq +++
(2) 22ap +
(3) 22bq + (4) 2222 abpq +−−
9. A rod AB of length 4 units moves horizontally with its left end A always on the circle x2 + y2 – 4x – 18y – 29 = 0. Then, the locus of the other end B is
(1) x2 + y2 – 12x – 8y + 3 = 0
(2) x2 + y2 – 12x – 18y + 3 = 0
(3) x2 + y2 – 4x – 8y – 29 = 0
(4) x2 + y2 – 4x – 16y + 19 = 0
10. If the area of the circle x2 + y2 + 4x + 2y + k = 0 is 5 p square cm, then k = _____.
11. The centroid of an equilateral triangle is (0,0) and the length of the altitude is 6. Then the radius of the circumcircle of the triangle is ____.
12. If the lines lx + 2y + 3 = 0 and 2y + mx +4 = 0 cut the coordinate axes in concyclic points, then lm = _____.
Answer Key
(1) 1 (2) 3 (3) 1 (4) 3
(5) 2 (6) 1 (7) 1 (8) 1
(9) 2 (10) 0 (11) 4 (12) 4
13.2 TANGENT AND NORMAL
Let P and Q be any two points on the circle. If point Q approaches P along the curve on either of the sides, then the limiting position of secant line PQ is called tangent to the circle at P.
1. If a line intersects the circle at two coincident points, then that secant is called tangent of circle at that coincident point.
2. Tangent line is perpendicular to the line joining the centre of the circle and the point of contact.
3. The condition that the line L = 0 is to be tangent to the circle S = 0 is that the perpendicular distance from the centre to the line L = 0 is equal to radius.
4. The point of contact of a tangent and circle is the foot of the perpendicular of centre of the circle on the tangent.
13.2.1 Position of a Point with Respect to Circle
The position of a point P(x1, y1) with respect to the circle S = 0:
i. If S11 > 0, then P lies outside the circle S = 0.
ii. If S11 = 0, then P lies on the circle S = 0.
iii. If S11 < 0, then P lies inside the circle S = 0.
1. Power of a point: The power of a point P(x1, y1) with respect to the circle having centre at C and radius r is (CP)2 – r2 = S11.
i. A point P lies inside the circle ⇔ power of P is negative.
ii. A point P lies outside the circle ⇔ power of P is positive.
iii. A point P lies on the circle ⇔ power of P is zero.
2. If a line passing through a point P( x1, y1) intersects the circle at two points A and B, then PAPB = |S11|
3. Minimum and maximum distance from a point to the circle: Let P be any point in the plane of circle S = 0 having centre at C and radius r. Then,
• shortest distance from P to the circle is |CP – r|
• longest distance from P to the circle is CP + r
If the line passing through the point P intersects the circle S = 0 at two points A and B, and suppose A is nearest point to P and B is farthest point, then
• the point A divides the line segment CP in the ratio r : CP – r.
• the point B divides the line segment CP in the ratio r : CP + r externally.
10 If the line P(–2, 3) meets the circle x2 + y2 –4x + 2y + k = 0 at A and B such that PAPB = 31, then find the radius of the circle.
Sol: Given: Circle S ≡ x2 + y2 – 4x + 2y + k = 0 and the point P = (–2, 3)
Given that PAPB = 31 ⇒ S11 = 31
⇒ 4 + 9 + 8 + 6 + k = 31
⇒ k = 4
Now, 4141 r =+−=
Try yourself:
9. For the circle x2 + y2 + 6x + 8y – 96 = 0, find
a. position of P(1, 2) with respect to the circle
b. power of P(1, 2) with respect to the circle
b. Power of the point = –69
Ans: a. P lies inside the circle.
13.2.2 Position of a Line with Respect to the Circle
Let S = 0 be a circle and L = 0 be a line; the position of line L = 0 with respect to the circle S = 0 is in three ways. Let r be the radius of the circle and d be perpendicular distance from the centre of the circle to the line.
1. If r > d, then the line L = 0 intersects the circle S = 0 at two points. In this case, the length of the chord is 22 2 rd .
S = 0
L = 0
S = 0
L = 0
S = 0
2. If r = d , then the line L = 0 touches the circle S = 0 at a point. C
L = 0
2. If r < d , then the line L = 0 does not intersect the circle S = 0.
3. Image of a circle with respect to line: Radii of circle and its image circle are equal.
S = 0
L = 0
C' S' = 0
Centre of image circle S' = 0 is image of centre C of the circle S = 0 with respect to the line L = 0.
4. Consider a circle S = 0 having centre at C and radius r and the line L = 0 is away from the circle S = 0. d is perpendicular distance from C to the line L = 0.
i. The minimum distance from S = 0 to L = 0 is d–r
ii. The maximum distance from S = 0 to L = 0 is d+r
5. If three lines form a triangle, then the number of circles that touch all the three lines is 4.
6. If two of three lines are parallel, then the number of circles that touch all three lines is 2.
7. If all the three lines are parallel, then there is no circle which touches all the lines.
8. If the three lines are concurrent, then the number of circles that touch all the three lines is 1, and it is a point circle.
11 Find the equation of circle having centre at (1, 2) and touching the line 3 x – 4y + 1 = 0.
Sol: Given: The centre of the circle is C(1, 2).
Since the line 3 x – 4 y + 1 = 0 touches the circle, the perpendicular distance from C(1, 2) to the line 3x – 4y + 1 = 0 is radius of the circle.
3(1)4(2)1381
Hence, the equation of the circle is
Therefore, equation of circle is 25(x2 + y2) – 50x – 100y + 84 = 0.
Try yourself:
10. If d 1 and d 2 are the longest and shortest distances of (–7, 2) from any point (a,b) on the curve whose equation is x2 + y2 – 10x – 14y = 51, then GM of d1 and d2. Ans: 211
13.2.3 Equation of Tangent and Normal
1. Equation of tangent at P(x1, y1) to the circle S = 0 is S1 = 0.
2. Equation of tangent at P(x1, y1) to the circle x2 + y2 = r2 is xx1 + yy1 – r2 = 0.
3. Equation of tangent at q to the circle x2 + y2 = r2 is x cos q + y sin q = r.
4. Equation of circle which touches the line L = 0 at a point P(x1, y1) can be taken as (x – x1)2 + (y – y1)2 + λ L = 0, where λ is parameter.
Normal: A line passing through the point P and perpendicular to the tangent at P to the circle is called normal to the circle at P
5. Normal always passes through the centre of the circle.
6. Equation of normal at P(x1, y1) to the circle x2 + y2 = r2 is xy1 – x1y = 0.
7. Equation of normal at q to the circle x2 + y2 = r2 is x sin q – y cos q = 0.
8. The condition of line L = 0 to be normal to the circle S = 0 is that the line L = 0 must pass through the centre of the circle.
9. Equation of normal to the circle S = 0 at P(x1, y1) is (y + f)(x – x1) = (x1 + g)(y – y1).
12 Find the point of contact of the circle x 2 + y 2 – 6 x + 4 y – 12 = 0 and the line 4x – 3y + 7 = 0.
Sol: Centre of the circle is (3, –2).
The point of contact is foot of the perpendicular from (3, –2) on the line 4 x – 3y + 7 = 0.
Hence, (()()) 43327 32 43169 32 1 43 hk hk −+−−−+ == −+ −+ ==−
It implies that h – 3 = –4 ⇒ h = –1 and k + 2 = 3 ⇒ k = 1.
Therefore, the point of contact is (–1, 1).
Try yourself:
11. Find the equation of tangent and normal at (1, 1) to the circle 2x2 + 2y2 – 2x – 5y + 3 = 0. Ans: 2x – y = 1, x + 2y = 3
13.2.4 Condition for Tangency
1. If the perpendicular distance from the centre to the line is equal to the radius of the circle, then that line is called tangent to the circle.
2. The condition that the line y = mx + c is to be tangent to the circle x2 + y2 = a2 is c2 = a2 (1 + m2) and the point of contact is 22 , ama cc
3. The condition that the line lx + my + n = 0 is to be tangent to the circle x 2 + y 2 = a 2 is n 2 = a 2 ( l 2 + m 2 ) and the point of contact is 22 , alam nn
4. Slope form of tangent : The slope form of tangent to the circle
x2 + y2 = a2 is 2 1 ymxam =±+ and the point of contact is 22 , ama cc .
5. The slope form of tangent to the circle
x2 + y2 + 2gx + 2fy + c = 0 is () 2 1 yfmxgrm +=+±+ where r is radius of the circle.
6. For the circle x2 + y2 + 2gx + 2fy + c = 0,
i. equation of tangent parallel to x-axis is y + f = ± r
ii. equation of tangent parallel to y-axis is x + g = ± r
iii. equation of tangent parallel to the line lx + my + n = 0 is 22 g lxmylmfrlm +++=±+ , where r is the radius of the circle.
13. Find the equations of tangents to the circle
x2 + y2 – 4x – 6y + 3 = 0, which are inclined at 45° with x-axis.
Sol: Centre of the circle is (2, 3) and radius is 10 r =
The slope form of a tangent to the given circle is () 3121011
220 201 yx x xy −=−±+ =−± −=±−
Therefore, the equations of tangents are 201 xy−=±−
Try yourself:
12. Find the equation of tangent to the circle
x2 + y2 – 4x + 6y – 12 = 0 which is parallel to the line x + y – 8 = 0.
Ans: 521 xy+=±−
13.2.5 Position of a Circle with Respect to Axes
Position of a circle S = 0 with respect to x - axis:
1. The circle x2 + y2 + 2gx + 2fy + c = 0
• intersects the x-axis, if g2 > c
• touches the x-axis, if g2 = c
• does not intersect the x-axis, if g2 < c.
2. The length of x–intercept made by the circle
x 2 + y 2 + 2 gx + 2 fy + c = 0 on x- axis is
2 2 gc
3. The equation of a circle which touches x-axis and has centre at (a, b) can be taken as x 2 + y 2 – 2 ax – 2 by + a 2 = 0. In this case, absolute value of ordinate of centre is equal to radius.This circle touches x -axis from bottom if f < 0 and it touches the x -axis from top if f > 0.
Position of a circle S = 0 with respect to y-axis:
4. The circle S ≡ x2 + y2 + 2gx + 2fy + c = 0
• intersects the y-axis if f 2 > c
• touches the y-axis if f 2 = c
• does not intersect the y-axis if f 2 < c
5. The length of y-intercept made by the circle x 2 + y 2 + 2 gx + 2 fy + c = 0 on y -axis is
2 2 fc
6. The equation of a circle which touches y-axis and has centre at (a, b) can be taken as x2 + y2 – 2ax – 2by + b2 = 0. In this case, the absolute value of abscissa of centre is equal to radius. This circle touches y-axis from left if g < 0 and it touches the y-axis from right if g > 0.
Position of circle with respect to both axes:
7. The circle S ≡ x2 + y2 + 2gx + 2fy + c = 0
• intersects both axes ⇔ g2 > c, f 2 > c
• touches both axes ⇔ g2 = f 2 = c
• does not intersect both axes ⇔ g2 < c, f 2 < c
8. Equation of a circle that touches both axes and lies in the first quadrant can be taken as x2 + y2 – 2kx – 2ky + k2 = 0
O y x (k, k) k k
9. Equation of a circle that touches both axes and lies in the second quadrant can be taken as x2 + y2 + 2kx – 2ky + k2 = 0.
O y x
10. Equation of a circle that touches both axes and lies in the third quadrant can be taken as x2 + y2 + 2kx + 2ky + k2 = 0.
O y x
11. Equation of a circle that touches both axes and lies in the fourth quadrant can be taken as x2 + y2 – 2kx + 2ky + k2 = 0
O y x
14 Find the equation of a circle which passes through (3, 6) and touches both axes.
Sol: Since the point (3, 6) lies in the first quadrant, the circle must lie in the first quadrant.
Equation of any circle which touches both the axes and lies in the first quadrant can be taken as x2 + y2 – 2kx – 2ky + k2 = 0.
Since it is passing through (3, 6), then 9 + 36 –6k – 12k + k2 = 0
It implies that k = 3, k = 15.
Therefore, the equation of the circle is x2 + y2 –6x – 6y + 9 = 0 or x2 + y2 –30x – 30y + 225 = 0
Try yourself:
13. Find the equation of a circle passing through the point (–1, 0) and touching y -axis at (0, 2).
Ans: x2 + y2 + 5x – 4y + 4 = 0
TEST YOURSELF
1. If a chord through P cuts the circle x2 + y2 + 2gx + 2fy + c = 0 at A and B, and another chord through P cuts the circle at C and D, then
(1) PA.PB<PC.PD
(2) PA.PB=PC.PD
(3) PA.PC=PB.PD
(4) PA . PB > PC . PD
2. The equation of the circle with centre (4, 1) and having 3x + 4y – 1 = 0 as tangent is (1) x2 + y2 – 8x= 0 (2) x2 + y2 – 8x–2y+ 8 = 0 (3) x2 + y2 – 8x+2y+ 8 = 0 (4) x2 + y2 – 8x+ 4 = 0
3. If the tangent to the circle x2 + y2 – 2x– 4y + 4 = 0 at (1, 1) is kx – y + 1 = 0, then k = (1) –1 (2) 0 (3) 1 (4) 2
4. The equations of the tangents to the circle x2 + y2 = 25 with slope 2 is
(1) 25yx=± (2) 225yx=±
(3) 235yx=± (4) 255yx=±
5. The point of contact of 32 yx=+ with x2 + y2 = 9 is
(1) 33 , 22
(3) 33 , 22
(2)
(4)
32 , 22
33 , 22
6. The equations of circles which touch both the axes, whose centres are at a distance of 22 units from the origin, are
(1) x2 + y2 ± 4x ± 4y + 4 = 0
(2) x2 + y2 ± 2x ± 2y + 4 = 0
(3) x2 + y2 ± x ± y + 4 = 0
(4) x2 + y2 ± 3x ± 3y + 4 = 0
7. The equation of the normal to the circle x2 + y2 + 6x + 4y – 3 = 0 at (1, –2) is (1) y + 1 = 0 (2) y + 2 = 0 (3) y + 3 = 0 (4) y – 2 = 0
8. If a chord of circle x2 + y2 = 8 makes equal intercepts of length ‘a' on the coordinate axes, then |a| < (1) 2 (2) 4 (3) 22 (4) 8
9. Number of circles touching all the lines x + 4y + 1 = 0, 2x + 3y +3=0, and x – 6y + 3 = 0 is (1) 0 (2) 2 (3) 4 (4) infinite
10. If the line through P(8, 3) meets the circle x2 + y2 – 8x– 10y+ 26 = 0 at A and B, then PA.PB = ______.
11. The intercept made by the circle x2 + y2 – 6x + 2y – 28 = 0 on the line 2x – 5y + 18 = 0 is equal to ______.
12. The number of integral values of λ so that x – 2y + λ = 0 intersects the circle x2 + y2 –6x+ 1 = 0 in two distinct points is ____.
The number of tangents drawn from an external point to the circle S = 0 is 2. Number of tangents drawn from a point on the circle S = 0 is 1. Number of real tangents drawn from an internal point to a circle S = 0 is 0.
13.3.1 Length of the Tangent
If the tangent drawn from an external point P to the circle S = 0 touches the circle at Q, then PQ is called the length of the tangent from P to the circle S = 0.
1. The length of the tangent to the circle S = 0 from the point P( x1, y1) is 11S .
2. If the point P lies outside the circle, then square of the length of the tangent drawn from P to the circle S = 0 is equal to the power of the point P with respect to the circle S = 0.
15. If the length of the tangent from (2, 5) to the circle x2 + y2 – 5x + 4y + k = 0 is 37 , then find the value of k.
Sol: Use the formula for the length of the tangent.
Given, 11 11 3737 425102037 3937 2 SS k k k =⇒= ⇒+−++= ⇒+= ⇒=−
Try yourself:
14. Find the locus of a point P which is moving such that lengths of tangents drawn from P to circles x2 + y2 – 4x – 6y – 12 = 0 and x2 + y2 + 6x + 18y + 26 = 0 are in the ratio 1 : 1.
Ans: 5x + 12y + 19 = 0
13.3.2 Equation of Pair of Tangents
1. The combined equation of a pair of tangents drawn from the point P( x1, y1) to the circle S = 0 is S 1 2 = SS11.
2. If m 1 and m 2 are the slopes of tangents drawn from the external point P( x 1 , y 1 ) to the circle x 2 + y 2 = a 2 , then 11 12 22 1 2 xy mm xa += and 22 1 12 22 1 . ya mm xa =
3. If q is the angle between the pair of tangents drawn from a point P( x1, y1) to the circle S = 0, then 11 tan 2 r S θ
4. If the tangents are perpendicular to each other, then S11 = r2
5. Director cirle: The locus of a point, from which the tangents drawn to the circle S = 0 are perpendicular, is a circle called director circle.
• Director circle of circle S = 0 is concentric with S = 0 and having radius 2, r where r is the radius of the circle S = 0.
• The locus of point of intersection of perpendicular tangents to the circle
a) x2 + y2 = r2 is x2 + y2 = 2r2
b) (x – x1)2 + (y – y1)2 = r2 is (x – x1)2 + (y – y1)2 = 2r2
6. The locus of point of intersection of perpendicular tangents drawn one to each of the circles
•x 2 + y 2 = a 2 and x 2 + y 2 = b 2 is x2 + y2 = a2 +b2
•x 2 + y 2 + 2 gx + 2 fy + c = 0 and x 2 + y 2 + 2 gx + 2 fy + c ' = 0 is x2 + y2 + 2gx + 2fy + c = g2 + f2 – c '
7. The locus of point of intersection of two tangents which include an angle q with respect to the circle x2 + y2 = a2 is x2 + y2
22 cosec 2 a = θ .
8. If PA, PB are two tangents drawn from the point P to the circle S = 0 having centre at C and radius r, then
• the circumcentre of triangle PAB is midpoint of CP and circumradius is 2 CP
• the area of the triangle PAC is 11 1 2 rS ; it is a right-angled triangle
16. If two perpendicular tangents can be drawn from (0, –3) to circle x 2 + y 2 – 2 ax – 8 y + 15 = 0, then find the value of [|a|], where [.] denotes greatest integer function.
Sol: Given: Circle S ≡ x2 + y2 – 2ax – 8y + 15 = 0 and Sol.the given point P(0, –3).
Since tangents drawn from P to the circle S = 0 are perpendicular, then 11rS =
⇒ r2 = S11
⇒ a2 + 16 – 15 = 0 + 9 – 0 – 8(–3) + 15
⇒ a2 + 1 = 9 + 24 + 15
⇒ a2 + 1 = 48
⇒ a2 = 47 47 a ⇒=±
Now, [|a|] = 476=
Try yourself:
15. Find the locus of point of intersection of two tangents to the circles x2 + y2 = a2, given that the angle between the tangents is 2 . 3 π
Ans: 3(x2 + y)2 = 4a2
13.3.3 Chord of Contact
Definition: The chord of contact of an external point P to the circle S = 0 is the line joining the points of contact of a pair of tangents drawn from P to the circle S = 0.
Chord of contact of P
The equation of the chord of contact of P(x1, y1) with respect to the circle S = 0 is S 1 = 0. If the point P lies on the circle, then the chord of contact is tangent to the circle at P. If the point P lies inside the circle, then the chord of contact does not exist.
2. S r Sr +
Length of the chord of contact of the point P( x 1, y 1) with respect to the circle S = 0 is 11 2 11
• the area of the quadrilateral formed by the two tangents drawn from an external point to a circle and a pair of radii through their point of contact is 11rS
• S = 0 is a circle in standard form with centre C and radius r. If P( x 1 , y 1 ) is a point then the area of the triangle formed by pair of tangents from P and chord of contact of P is () 3 2 11 2 11 rS Sr +
17. Find the length of the chord of the point (3, 6) with respect to the circle x2 + y2 = 10.
Sol: Given: Circle S ≡ x2 + y2 = 10,where r2 = 10.
Let P = (3, 6), S11 = 35
Length of chord of contact = 11 2 11 2. S r Sr + 35270 210 453 ==
Try yourself:
16. Find the equation of chord of contact of the tangents are drawn from the point (1, 4) to the circle x2 + y2 + 8x + 12y + 6 = 0.
Ans: 5x + 10y + 34 = 0
13.3.4 Midpoint of Chord
Equation of the chord of the circle S = 0 whose midpoint (x1, y1) is S1 = S11
1. If P(x1, y1) is midpoint of the chord, then the length of chord is 11 2 S .
2. The locus of midpoints of chords of circle x2 + y2 = a2, which subtends an angle q at centre, is x2 + y2 = r2 cos2 q /2.
3. The locus of midpoints of chords of circle x2 + y2 = a2 of length 2k units is x2 + y2 = a2 – k2
18. If (3, –2) is the midpoint of the chord AB of the circle x2 + y2 – 4x + 6y – 5 = 0, then find AB
Sol: Given: Circle S ≡ x2 + y2 – 4x + 6y – 5 = 0, P = (3,–2)
11 2294121258ABS
Try yourself:
17. Find the locus of midpoints of the chords of the circle 4x2 + 4y2 – 12x + 4y + 1 = 0 that subtends an angle of 2π/3 at its centre. Ans: 22 31 30 16 xyxy+−++=
TEST YOURSELF
1. The length of the tangent from (1, 1) to the circle 2x2 + 2y2 + 5x+ 3y+ 1 = 0 is
(1) 13 2 (2) 3 (3) 2 (4) 1
2. Locus of the point of intersection of perpendicular tangents drawn one to each of the circles x2 + y2 = 8 and x2 + y2 = 12 is
(1) x2 + y2 = 4 (2) x2 + y2 = 20
(3) x2 + y2 = 208 (4) x2 + y2 = 16
3. The angle between the tangents drawn from (0, 0) to the circle x2 + y2 + 4x – 6y + 4 = 0 is
(1) 1 5 sin 13 (2) 1 5 sin 12
(3) 1 12 sin 13 (4) 2 π
4. The pair of tangents from origin to the circle x2 + y2 + 4x + 2y + 3 = 0 is
(1) (2x + y)2 = 3(x2 + y2)
(2) (4x + 2y)2 = 3(x2 + y2)
(3) (2x + y)2 = 3(x2 + y2)
(4) not existing
5. Tangents PA and PB are drawn to x2 + y2 = 9 from any arbitrary point P on the line x + y = 25. The locus of the midpoint of chord AB is
(1) 25(x2 + y2) = 9(x + y)
(2) 25(x2 + y2) = 3(x + y)
(3) 5(x2 + y2) = 3(x + y)
(4) none of these
6. The locus of the midpoints of the chords of the circle x2 + y2 − ax − by = 0, which subtends a right angle at , 22 ab , is
(1) ax + by = 0
(2) ax + by = a2 + b2
(3) 22 22 0 8 xyaxbyab + +−−−=
(4) 22 22 0 8 xyaxbyab + +−−+=
7. The locus of the point ( f, g) such that the length of the tangent from ( f, g) to x2 + y2 = 6 is twice the length of the tangent from the same point to x2 + y2 + 3x + 3y = 0 is
(1) x2 + y2 + 4x + 4y + 2 = 0
(2) x2 + y2 + 4x – 4y + 2 = 0
(3) x2 + y2 – 4x + 4y + 2 = 0
(4) x2+y2–4x–4y+2=0
8. The range of values of λ(λ > 0) such that the angle θ between the pair of tangents drawn from (λ, 0) to the circle x2 + y2 = 4 lies in 2 , 23 ππ
is
(1) 4 ,22 3
(2) () 0,2 (3) (1,2) (4) (1,-1)
9. The area of the triangle (in sq. units) formed by the tangents drawn from P(4,4) to the circle S = x2 + y2 − 2x − 2y − 7 = 0 and the chord of contact of P with respect to S = 0 is (1) 4.5 (2) 5.4 (3) 6.75 (4) 1.5
10. Angle between tangents drawn from a point P to circle x2 + y2 −4x−8y+8=0 is 60 °. Then the length of chord of contact of P is .
11. The square of length of the tangent from a point on x2 + y2 + 4x + 8y – 4 = 0 to 2x2 +2y2 + 8x + 16y + 1 = 0.
12. The area of the quadrilateral formed by a pair of tangents from the point (4, 5) to the circle x2+ y2 – 4x – 2y – 11 = 0 and a pair of its radii is ____
Determining the relative location of two circles requires knowledge of the geometric connections that their common tangents generate. Determining if a circle is nontangent, internally tangent, or externally tangent requires knowledge of shared tangents. The closeness and spatial layout of the circles in a particular configuration may be understood by analysing common tangents.
13.4.1 Common Tangents
Definition: A line L is said to be a common tangent to the circles S = 0 and S' = 0, if it is a tangent to both the circles.
There are two types of common tangents:
Direct common tangents: Let L = 0 be a common tangent to the circles S = 0 and S' = 0. If the circles S = 0 and S' = 0 lie on the same side of the line L = 0, then the line L = 0 is called direct common tangent.
L = 0
S' = 0
S = 0
Transverse common tangent: Let L = 0 be a common tangent to the circles S = 0 and S' = 0. If the circles S = 0 and S' = 0 lie on either side of the line L = 0, then the line L = 0 is called transverse common tangent.
S = 0
L = 0
S' = 0
In general, there exists at most two direct common tangents and two transverse common tangents to a given pair of non-concentric circles. There is no common tangent to the concentric circles.
Length of common tangents: If d is the distance between the centres of two circles whose radii are r1 and r2, then
• the length of the direct common tangents is ()2 2 12drr
• the length of the transverse common tangents is ()2 2 12drr −+
Centres of similitude: Let S = 0 and S' = 0 be two circles with centres C1 and C2 and radii r1 and r2, respectively.
1. The point of intersection of direct common tangent is called external centre of similitude (A 2). This point divides the line joining the centres C1 and C2 in the ratio r1 : r2 externally.
2. The point of intersection of transverse common tangents is called internal centre of similitude (A 1). This point divides the line joining the centres C1 and C2 in the ratio r1 : r2 internally.
13.4.2 Number of Common Tangents
Let S = 0, S' = 0 be any two circles having centres at C1 and C2 and radii r1, r2. Suppose, C1C2 = d.
1. If d > r1 + r2, then one circle completely lies outside of the other, and the circles do not intersect each other.
i. In this case, there will four common tangents; 2 are direct common tangents and 2 are transverse common tangents, as shown below. P1 C1 C2 P2 Q Q2 Q1
ii. In this case, the minimum distance between the two circles is d – (r1 + r2) and the maximum distance between the circles is d + (r1 + r2).
19 If the distance between the cerntres of two circles of radii 3 and 4 is 25, then find the length of transverse common tangent.
Sol: Given: d = 25, r1 = 3, r2 = 4
Now, length of transverse common tangent ()2 2 22 12 25(34)57624 drr−+=−+==
Try yourself:
18. Find the external centre of similitude of the circles x2 + y2 – 2x – 6y + 9 = 0 and x2 + y2 = 4.
Ans: (2, 6)
2. If d = r1 + r2, then one circle is completely touching the other circle externally. In this case, there will be three common tangents; 2 direct common tangents and 1 transverse common tangent, as shown below. P1 C1 C2 P2 Q L Q2 Q1
3. If d = |r1 – r2|, then one circle is touching the other circle internally. In this case, there will be only one common tangent, that is direct common tangent, as shown below.
C1 P C2
4. If |r1 – r2| < d < r1 + r2, then both circles intersect at two points. In this case, there will be two common tangents, and those two are direct common tangents, as shown below.
TEST YOURSELF
1. The circles x2 + y2 – 8x + 6y + 21 = 0 and x2 + y2 + 4x – 10y – 115 = 0 are (1) intersecting
(2) touching externally
(3) touching internally
(4) one is lying inside the other
2. The circles x2 + y2 – 12x + 8y + 48 = 0 and x2 + y2 + 4x – 10y – 115 = 0 are
5. If 0 < d < |r1 – r2|, then one circle completely lies inside the other. In this case, there will be no common tangent.
6. If d = 0, r1 ≠ r2, then both circles are concentric circles with different radii. In this case, there will be no common tangent.
7. If d = 0, r1 = r2, then both circles are coincident circles. In this case, there will be an infinite number of common tangents.
S.No
20 Discuss the relative position of two circles, x2 + y2 + 6x + 6y + 14 = 0 and x2 + y2 – 2x – 4y – 4 = 0, and find the number of possible common tangents that exist.
Sol: C1(–3,–3), r1 = 2, C2 = (1, 2), r2 = 3
Distance between the points 12 162541 CC =+=
Observe that C1C2 > r1 + r2
So, both circles do not intersect each other; one circle completely lies outside the other. Hence, the number of possible common tangents is 4.
Try yourself:
19. Prove that the circles x2 + y2 – 8x – 6y + 21 = 0 and x2 + y2 – 2y –15 = 0 have exactly two common tangents. Also find the intersection of those tangents.
Ans: (8, 5)
(1) intersecting
(2) touching externally
(3) touching internally
(4) one is lying inside the other
3. The number of common tangents to x2 + y2 = 256 and (x – 3)2 + (y – 4)2 = 121 is (1) one (2) two (3) four (4) zero
4. The internal centre of similitude of the circles x2 + y2 – 2x + 4y + 4 = 0 and x2 + y2 + 4x – 2y + 1 = 0 divides the segment joining their centres in the ratio
(1) 1 : 2 (2) 2 : 1
(3) –1 : 2 (4) –2 : 1
5. The two circles x 2 + y 2 = ax and x 2 + y 2 = c 2 (c > 0 ) touch each other if (1) |a|=c (2) a = 2c (3) | a | = 2c (4) 2 | a | = c
6. The number of common tangents to the circles x2 + y2 – 4x – 6y – 12 = 0 and x2 + y2 + 6x + 18y + 26 = 0 is
(1) 2 (2) 3 (3) 4 (4) 1
7. The centres of those circles that touch the circle x2+y2–8x – 8y – 4 = 0 externally and also touch the x-axis lie on
(1) an ellipse which is not a circle
(2) a hyperbola
(3) a parabola
(4) a circle
8. If the two circles (x – 1)2+(y – 3)2 = r2 and x 2 + y 2 –8 x + 2 y + 8 = 0 intersect at two distinct points, then (1) r > 2 (2) 2 < r < 8 (3) r < 2 (4) r = 2
9. If 1 ,1 3 is a centre of similitude for the circles x2 + y2 = 1 and x2 + y2 − 2x − 6y − 6 = 0, then the length of common tangent of the circles is (1) 1 3 (2) 4 3 (3) 1 (4) cannot be determined
10. If the curves x2 − 6x + y2 + 8 = 0 and x2 − 8y + y2 + 16 − k = 0, k > 0, touch each other at a point, then the largest value of k is ____.
11. The minimum distance between any two points P 1 and P 2, while considering that point P1 is on one circle and P2 is on the other circle, for the given circle equations x2 + y2 − 10x − 10y + 41 = 0 and x2 + y2−24x −10y +160 = 0, is _________.
12. Two circles, each of radius 5 units, touch each other at the point (1, 2). If the equation of their common tangent is 4x + 3y = 10, and C 1 (α, β) and C 2 (γ, δ), C 1 ≠ C 2 are their centres, then |(α + β)(γ + δ)| is equal to __________.
A family of circles is a group of circles that share certain common properties or relationships often expressed through mathematical equations or geometric characterstics. Understanding the concepts associated with a family of circles involves exploring the angle between two intersecting circles, the radical centre and radical axes, and the coaxial system.
13.5.1 Angle between Two Intersecting Circles
Angle between two intersecting circles is defined as the angle between the tangents drawn at their point of intersection.
1. If two circles intersect at two points, then the angle between the circles at those two points are the same.
2. If q is the angle between two circles S = 0 and S' = 0, having centres at C 1 and C2 and radii r 1 and r 2 , and C 1 C 2 = d , then 222 12 12 cos 2 drr rr
3. If q is an angle between two circles S ≡ x 2 + y 2 + 2 gx+ 2 fy + c = 0 and S' ≡ x 2 + y 2 + 2 g'x+ 2 f'y + c' = 0, then
4. If the angle between two circles is a right angle, then the two circles are said to be orthogonal.
5. If d is the distance between the centres of two orthogonal circles having radii r 1 and r 2 , then d2 = r1 2 + r2 2
6. The condition that two circles x2 + y2 + 2gx + 2fy + c = 0 and x2 + y2 + 2g'x+ 2f'y + c' = 0 may cut each other orthogonally is 2gg'+ 2ff' = c+c'
7. If two circles intersect orthogonally, then the tangent of one circle at their point of intersection is normal to the other circle.
21. Find the angle between the circles given by the equation x2 + y2 + 6x – 10y – 135 = 0 and x2 + y2 + – 4x + 14y – 116 = 0.
Sol: Given circles are x2 + y2 + 6x – 10y – 135 = 0 and x 2 + y 2 + – 4 x + 14 y – 116 = 0, where g = 3, f =–5, c = –135, g' = –2, f '= 7, c' = –116 Now, θ
+−− = ′′′+−+− 2222
22 cos 2 ccggff gfcgfc
1351162(3)(2)2(5)(7)
22 Find the value of k if the circles x2 + y2 – 5x – 14y – 34 = 0 and x2 + y2 + 2x + 4y + k = 0 are orthogonal to each other.
Sol: Use the condition for orthogonality, (2gg'+ 2ff') = c+c'
Hence, ()() 514734 52834 3334 1 k k k k
Therefore, k = 1
Try yourself:
20. Find the equation of the circle which passes through the point (0, –3) and intersects the circles given by the equations x 2 + y 2 – x – 7 y = 0 and x2 + y2 – 6x + 3y+ 5 = 0 orthogonally.
Ans: 3(x2 + y)2 + 2x + 4y + 15 = 0
13.5.2 Common Chord and Its Length
If two circles intersect at two points, the line joining those two points is called common chord of circles.
1. The equation of common chord of two circles S= 0 and S' = 0 is S – S' = 0.
2. The line S – S' = 0 is a common tangent to the circles if both the circles touch each other.
3. The length of the common chord is equal to the distance between the points of intersection of two circles.
5. If the angle between two circles is 90°, the length of common chord of those two circles is 12 22 12 2rr rr +
6. The length of the common chord of two circles is maximum when it is the diameter of the smaller circle.
7. The length of the common chord is zero, if two circles touch each other.
8. The condition that one circle S = 0 bisects the circumference of S' = 0 is that the common chord passes through the centre of S' = 0.
23. Show that the common chord of the circles x2 + y2 – 6x – 4y + 9 = 0 and x2 + y2 – 8x – 6y + 23 = 0 is the diameter of the second circle, and then find its length.
Sol: The common chord of the circles is S – S' = 0. It implies that x + y – 7 = 0.
It is passing through the centre (3, 4) of the second circle, so that the common chord of the given two circles is the diameter of the second circle.
The length of the common chord is diameter of the second circle, and it is equal to 229162322dr==+−=
Try yourself:
21. If the circle x2 + y2 + 4x + 22y + l = 0 bisects the circumference of the circle x2 + y2 – 2x + 8y – m = 0, then find the value of l + m.
13.5.3 Family of Circles
Ans: 50
Let S = 0 be a circle, and L = 0 be a line which intersects the circle at two distinct points. Then, equation of any circle passing through the points of intersection of S = 0 and L = 0 is S + λ L = 0.
4. If d is the distance between the centres of two circles, whose radii are r 1 and r 2, then the length of the common chord is 12 22 1212 2sin , 2cos rr rrrr θ ++θ where q is the angle between the intersecting circles.
= 0 S + λL = 0
Let S = 0 and S' = 0 be any two circles which intersect at two distinct points. Then, the equation of any circle passing through those two points is S + λ L = 0, where S – S' = L. S = 0 S' = 0 S + λL = 0 L = 0
2. Radical axis of two circles is nearer to the smallest circle among them.
3. Radical axis of two circles may not pass through the midpoint of the centres of the circles.
4. Radical axis of two circles is perpendicular to their line of centres.
5. Radical axis of two intersecting circles is common chord of those two circles.
6. Radical axis of two circles touching each other is common tangent at their point of contact.
7. Radical axis of two circles bisects the common tangent of the circle.
8. The centre of the circle which cuts the two circles orthogonally lies on the radical axes of those circles.
Let S = 0 and S' = 0 be any two circles that touch each other and their common tangent at their point of contact be L = 0. Then, the equation of the family of circles which touches both circles at the same point of contact is S + λ L = 0.
The equation of any circle passing through two points can be taken as S + λ L = 0, where S = 0 is a circle having those two points as ends of its diameter, and the line L = 0 is a line passing through the given two points.
Radical axis: The locus of a point, whose powers with respect to two circles S = 0 and S' = 0 are equal , is a straight line and it is called radical axis.
Equation of radical axis: The equation of radical axis of two circles S = 0 and S' = 0 is S – S' = 0.
Here, equations of both circles must be in standard form.
Properties of radical axis:
1. If the circles are concentric, then radical axis does not exist.
9. The locus of centres of circles, which intersects two circles orthogonally, is the radical axis.
10. A circle S = 0 is said to be bisecting the circumference of a circle S' = 0, if the radical axis of the two circles passes through the centre of the circle S' = 0.
11. If two circles touch each other externally, then the radical axis is a transverse common tangent.
12. If two circles intersect at two points, then the radical axis is the common chord of both circles.
13. If two circles touch each other internally, then the radical axis is a direct common tangent.
14. If one circle lies inside the other, then the radical axis lies away from the circles.
15. If one circle lies completely outside the other, then the radical axis lies between both the circles.The number of radical axes of n circles, of which the centres of no three circles are collinear, is 2nC .
16. The maximum number of radical axes of n circles is 2nC and the minimum number of radical axes is zero.
Radical centre: The point of concurrence of radical axes of three circles, whose centres are non-collinear taken in pairs, is called the radical centre of three circles.
S = 0 S' = 0
= 0
The powers of radical centre with respect to each of the three circles are equal. A circle having centre at radical centre and length of the tangent to any circle from radical centre as radius cuts the given three circles orthogonally.
■ The number of radical centres of n circles of which no three circles centres are collinear is nC3.
Coaxial system of circles: A system of circles, in which each pair of circles has the same radical axis, is called a coaxial system of circles. The centres of a coaxial system of circles are collinear.
If S = 0 is a circle and L = 0 is radical axis, then S + λ L = 0 represents the coaxial system of circles, in which S = 0 is a member and L = 0 is the radical axis.
24. Find the value of k, if the circle 3x2 + 3y2 + 10x + y – 27 = 0 bisects the circumference of the circle x2 + y2 = k.
Sol: Radical axis passes through the centre of the circle x2 + y2 = k.
Equation of radical axis is S – S' = 0
22 22 10 90 33 y xyxxyk +++−−−+= ⇒ 10x + y – 27 + 3k = 0
Since it passes through (0, 0), then 3k = 27 ⇒ k = 9
Try yourself:
22. Find the radical centre of the circles x 2 + y2 – x + 3y – 3 = 0, x2 +y2 – 2x + 2y – 2 = 0 and x2 + y2 + 2x + 3y – 9 = 0.
Ans: (2, –1)
TEST YOURSELF
1. The angle between the circles x2 + y2 + 4x + 8y +18 = 0 and x2 + y2 + 2x + 6y + 8 = 0 is (1) 4 π (2) 2 3 π (3) 6 π (4) 2 π
2. The angle between two circles, each passing through the centre of the other, is (1) 6 π (2) π 2 (3) 2 3 π (4) p
3. The circle passing through (1, 1) and cutting the two circles x2 + y2 – 4x – 2y– 4 = 0 and x2 + y2 – 2x – 4y– 4 = 0 orthogonally is
4. For the circles 3x2 + 3y2 + x + 2y– 1 = 0 and 2x2 +2 y2 + 2x – y– 1 = 0, radical axis is (1) 4x – 7y= 1
(2) 3x + y– 2 = 0
(3) x – 3y= 0
(4) 4x – 7y= 5
5. The length of the common chord of x2 + y2 + 2x + 3y+ 1 = 0 and x2 + y2 + 4x + 3y+ 2 = 0 is
(1) 2 (2) 2 (3) 22 (4) 4
6. Area of the triangle formed by the radical axis of the circles x 2 + y 2 = 4 and x2 + y2 + 2x + 4y– 6 = 0 with coordinate axes is (1) 1 4 sq. units (2) 1 3 sq. units (3) 1 6 sq. units (4) 1 8 sq. units
7. If the circle x2 + y2 + 8x – 4y+c = 0 touches the circle x2 + y2 + 2x + 4y– 11 = 0 externally and cuts the circle x2 + y2 – 6x + 8y+k = 0 orthogonally, then k = (1) 59 (2) –59 (3) 19 (4) –19
8. If P and Q are the points of intersection of the circles x2 + y2 + 3x + 7y+ 2p– 5 = 0 and x2 + y2 + 2x + 2y–p2 = 0, then there is a circle passing through P, Q and (1, 1) for (1) all except one value of P (2) all except two values of P (3) exactly one value of P (4) all values of P
CHAPTER REVIEW
Equation of Circle
1. The locus of a point in a plane which moves at a constant distance from the fixed point is called circle. The fixed point is called centre, and constant distance is called radius of the circle.
2. The equation of a circle having centre at (a, b) and radius r is (x – a)2 + (y – b)2 = r2 .
3. The general form of a circle is x 2 + y 2 + 2gx + 2fy+c = 0, having centre at (–g, –f) and radius 22 rgfc =+−
4. The conditions that the equation ax2 + 2hxy + by2 + 2gx + 2fy+c = 0 represents a circle are a = b, h = 0, and g2 + f2 – ac ≥ 0.
5. For the circle ax2 + ay2 + 2gx + 2fy+c = 0, centre is , gf aa and its radius is
9. The common chord of x2 + y2 – 4x– 4y = 0 and x2 + y2 = 16 subtends at the origin an angle equal to
10. The length of the common chord of the circles having radii 15 and 20, whose centres are 25 units apart is ____.
11. If the circumference of the circle x2 + y2 – 4x + 2y + a = 0 is bisected by the circle x2 + y2 + 2x – 3y + b = 0, then |b–a| = _____.
12. If the equation of a circle, which touches the line x + y = 5 at the point (–2, 7) and cuts the circle x 2 + y 2 + 4 x – 6 y + 9 = 0 orthogonally, is x2 + y2 + ax + by + c = 0, then a + 4b + c = _______.
6. If the radius of a circle is 1, then that circle is called unit circle.
7. If the radius of a circle is zero, then that circle is called zero circle.
8. If two or more circles have same centre, then those circles are called concentric circles.
9. The equation of concentric circle with x2 + y2 + 2gx + 2fy+c = 0 is x2 + y2 + 2gx + 2fy+k = 0.
10. Equation of the chord joining two points P(x1, y1) and Q(x2, y2) is S1 + S2 = S12
11. The length of the chord is 22 2 rd , where r is radius of the circle and d is perpendicular distance from the centre of the circle to the chord.
12. The foot of the perpendicular of centre of the circle on the chord is its midpoint.
13. The angle made by the chord at the centre of the circle is double the angle made by the chord at any point of major arc of the circle.
14. Position of a point: Let P (x1, y1) be any point in the plane of circle S = 0. Then,
• S11 = 0 ⇔ P lies on the circle
• S11 > 0 ⇔ P lies outside the circle
• S11 < 0 ⇔ P lies inside the circle
15. Power of a point: If C is the centre and r is the radius of a circle S = 0, then the power of the point P with respect to the circle S = 0 is CP2 – r2 If the secant line drawn from P(x1, y1) intersects the circle S = 0 at two points A and B, then PA . PB = S11; this is called power of the point P( x1, y1).
16. For the circle x2 + y2 + 2gx + 2fy+c = 0,
• x-intercept is 2 2 gc
• y-intercept is 2 2 fc
• x-axis is tangent ⇔ c = g2
• y-axis is tangent ⇔ c = f 2
• Both axes are tangents if and only if g2 = f 2 = c
Tangent and Normal
17. If a circle touches the x-axis, then its radius is equal to the absolute value of ordinate of the centre.
18. If a circle touches the y-axis, then its radius is equal to the absolute value of abscissa of the centre.
19. If a circle touches both the axes, then absolute values of both coordinates of centre are equal, and it is equal to radius.
20. Equation of tangent to the circle S = 0 at P(x1, y1) is S1 = 0.
21. Equation of tangent to the circle x2 + y2 = a2 at q is x cos q + y sin q= a
22. The condition for the line y = mx + c to touch the circle x2 + y2 = a2 is c2 = a2(1 + m2).
23. The condition for the line lx + my + n = 0 to touch the circle x 2 + y 2 = a 2 is n 2 = a2(l2 + m2), and the point of contact is 22 , lama nn
24. The condition for the line lx + my + n = 0 to touch the circle x2 + y2 + 2gx + 2fy+ c = 0 is (l2 + m2) (g2 + f2 – c) = (lg + mf –n)2, and the point of contact is the foot of the perpendicular from the centre to the tangent.
25. The slope form of tangent to the circle x2 + y2 = a2 is 2 1 ymxam =±+
26. For the circle (x – x1)2 + (y – y1)2 = a2 , (i) slope form of the tangent is () 2 11 1 yymxxam −=−±+
(ii) tangent parallel to x-axis is y = y 1 ± a
(iii) tangent parallel to y-axis is x = x 1 ± a
(iv) tangents parallel to lx+ my + n = 0 are () 22 11 lxmylxmyalm +=+±+
27. Normal at any point on the circle passes through the centre of the circle.
28. The equation of normal to the circle S = 0 at P(x1, y1) is the equation of line joining P(x1, y1) and the centre (– g, –f).
Pair of Tangents
29. The length of tangent to the circle S = 0 from P(x1, y1) is 11S
30. Two tangents can be drawn from an external point to a circle.
31. Equation of a pair of tangents from a point P(x1, y1) to the circle S = 0 is S 1 2 = SS11
32. If m1 and m2 are slopes of tangents from a point P(x1, y1) to the circle x2 + y2 = a2, then 11 12 22 1 2 xy mm xa += and 22 1 12 22 1 ya mm xa ⋅=
33. If q is the angle between the tangents, then 11 tan 2 r S θ
• the area of the quadrilateral formed by the two tangents drawn from an external point to a circle and a pair of radii through their point of contact is 11rS
, where r is radius of circle.
34. The locus of the point of intersection of perpendicular tangents to the circle S = 0 is called director circle; it is concentric with S = 0 and has radius 2r , where r is radius of S = 0. Equation of director circle of circle S = 0 is S = r2 .
(i) The locus of the point of intersection of perpendicular tangents to the circle
•x2 + y2 = r2 is x2 + y2 = 2r2
• ( x – x 1) 2 + ( y – y 1) 2 = r 2 is ( x – x 1) 2 + (y – y1)2 = 2r2
(ii) The locus of the point of intersection of perpendicular tangents drawn one to each of the circles
•x 2 + y 2 = a 2 , x 2 + y 2 = b 2 is x 2 + y 2 = a2 +b2
•x2 + y2 + 2gx + 2fy + c = 0, x2 + y2 +2gx + 2fy + c' = 0 is x2 + y2 + 2gx + 2fy + c
= g2 + f2 – c '
35. If PA and PB are two tangents to the circle S = 0 having centre at C and radius r, then circumcentre of triangle PAB is midpoint of CP and circumradius is 2 CP .
• Chord of contact of P is AB and its equation is S1 = 0
• Length of the chord AB is 11 2 11 2 S r Sr +
36. S = 0 is a circle in standard form with centre C and radius r. If P(x1,y1) is a point then the area of the triangle formed by pair of tangents from P and chord of contact of P is () 3 2 11 2 11 rS Sr +
37. Midpoint of chord: The equation of the chord having ( x 1, y 1) as its midpoint is S1 = S11; its length is 11 2 S
38. The locus of midpoints of chords of a circle x2 + y2 = a2, which subtends an angle q at the centre, is x2 + y2 = r2 cos2 q /2.
39. The locus of midpoints of chords of a circle x2 + y2 = a2 of length 2k units is x2 + y2 = a2 – k2 .
40. If both circles lie on the same side of common tangent, then that tangent is called direct common tangent and its length is ()2 2 12drr
41. If circles lie on the opposite sides of a common tangent, then that tangent is called a transverse common tangent and its length is ()2 2 12drr −+
42. Centres of similitude:
(i) The point of intersection of direct common tangents is called external centre of similitude, and this divides the line joining centres C1 and C 2 in the ratio r1 : r2 externally.
(ii) The point of intersection of transverse common tangents is called internal centre of similitude and this divides the line C1C2 in the ratio r1 : r2 internally.
Relative Position of Two Circles
43. Let S1 = 0 and S2 = 0 are two circles having centres at C 1 and C 2 and radii r 1 and r 2, respectively, and suppose, that d = C1C2
Condition Relative position Common tangents
d > r1 + r2
d = r1 + r2
| r1 – r2 | <
d < r1 + r2
One circle compleletly lies outside the other
Touch each other externally
Intersect at two points
d = |r1 – r2 | Two circles touch internally
d = 0, r1 ≠ r2
Direct common tangents –2
Transverse common tangens – 2
Direct common tangents –2
Transverse common tangens –1
Direct common tangents - 2
Transverse common tangens – 0
Direct common tangents – 1
Transverse common tangens – 0
Concentric circles No common tangents
d < |r1 – r2 | One circle completely lies inside the other No common tangents
d = 0, r1 = r2
Coincident circles Infinite direct common tangent
System of Circles
44. If q is the angle between two intersecting circles having centres at C 1 and C 2 and radii r 1 and r 2, then 222 12 12 cos 2 drr rr θ = , where d is the distance between the centres of the two circles.
45. If q is the angle between two circles S = 0 and S' = 0, then
46. If d is the distance between the centres of two orthogonal circles having radii r 1 and r 2 , then d2 = r1 2 + r2 2 .
47. The condition that two circles x 2 + y 2 + 2gx + 2fy + c = 0 and x2 + y2 + 2g'x + 2f'y + c' = 0 may cut each other orthogonally is 2gg' + 2ff' = c+c'.
48. If two circles intersect orthogonally, then the tangent of one circle at their point of intersection passes through the centre of the other circle.
49. The locus of a point whose powers with respect to two circles S = 0 and S' = 0 are equal is S – S' = 0; this is called radical axis of two circles. Equations of both circles must be in standard form.
• If the circles are concentric, then radical axis does not exist.
• Radical axis of two circles is perpendicular to their line of centres.
• Radical axis of two circles bisects the common tangent of the circle.
• The locus of centres of circles, which intersects two circles orthogonally, is the radical axis.
• A circle S = 0 is said to be bisecting the circumference of a circle S' = 0, if the radical axis of the two circles passes through the centre of the circle S' = 0.
• If two circles touch each other externally, then the radical axis is a transverse common tangent.
• If two circles intersect at two points, then the radical axis is the common chord of both circles.
• If two circles touch each other internally, then the radical axis is a direct common tangent.
• The number of radical axes of n circles, of which the centres of no three circles are collinear, is 2nC .
• The maximum number of radical axes of n circles is 2nC and the minimum number of radical axes is zero.
50. The equation of common chord of two intersecting circles S = 0 and S' = 0 is S – S' = 0.
51. Length of common chord is 12 22 1212 2sin 2cos rr rrrr θ ++θ , where q is the angle between two intersecting circles and r1 and r2 are the radii of the two circles.
52. If the angle between two circles is 90°, the length of common chord of those two circles is 12 22 12 2rr rr +
53. Family of circles:
(i) Let S = 0 be a circle and L = 0 be a line. The equation of any circle passing through the points of intersection of S = 0 and L = 0 can be taken as S + λL = 0.
(ii) Let S = 0 and S' = 0 be any two circles. The equation of any circle passing through the points of intersection of both circles S = 0 and S' = 0 can be taken as S + λ (S – S') = 0.
54. The point of concurrence of radical axes of three circles, whose centres are noncollinear, taken in pairs, is called the radical centre of the three circles. A system of circles in which each pair of circles has the same radical axis is called a coaxial system of circles.
• The number of radical centres of n circles of which no three circles centres are collinear is nC3.
Exercises
JEE MAIN LEVEL
Level – I
Equation of Circle
Single Option Correct MCQs
1. If one end of the diameter of the circle x2 + y2 – 6x + 4y – 12 = 0 is (7, –5), then the other end of the diameter is
(1) (–1, –3) (2) (–4, 3) (3) (–1, 1) (4) (–4, 4)
2. If the lines 3x – 4y – 7 = 0 and 2x – 3y – 5 = 0 are two diameters of a circle of area 49 p square units, then the equation of circle is
(1) x2 + y2 + 2x – 2y – 47 = 0
(2) x2 + y2 + 2x – 2y – 62 = 0
(3) x2 + y2 – 2x + 2y – 62 = 0
(4) x2 + y2 – 2x + 2y – 47 = 0
3. A square is inscribed in the circle x2 + y2 –4x +6 y – 5 = 0 whose sides are parallel to the co-ordinate axes then vertices of square are
4. A rectangle ABCD is inscribed in a circle with a diameter lying along the line 3y = x + 10. If A = (–6, 7), B = (4, 7), then the area of the rectangle is
(1) 80 sq. units (2) 40 sq. units (3) 160 sq. units (4) 20 sq. units
5. The length of chord intercepted by the circle x2 + y2 + 2x + 4y – 20 = 0 on the line 3x + 4y – 6 = 0 is
(1) 521 (2) 421 5
(3) 821 5 (4) 52
6. If the chord y = mx + 1 of the circle x2 + y2 = 1 subtends an angle of measure 45° at the major segment of the circle, then m = (1) 2 (2) –1 (3) –2 (4) 3
7. A square is inscribed in the circle x 2 + y 2 – 2 x + 8y – 8 = 0 whose diagonals are parallel to axes and a vertex in the first quadrant is A then OA is
(1) 1 (2) 2
(3) 22 (4) 3
8 If 1 , i i m m , i = 1, 2, 3, 4 are concyclic points then the value of m1 m2 m3 m4 is (1) 1 (2) –1 (3) 0 (4) ∞
Numerical Value Questions
9. If a chord of the circle x2 + y2 – 4x – 2y – c = 0 is trisected at the points (1/3, 1/3) and (8/3, 8/3), then the radius of the circle will be______.
10. A circle of constant radius 3k passes through origin and meets coordiante axes at A and B, then locus of centroid of triangle OAB is x2 + y2 = pk2 then the value of p = _____.
11. If π/6 and π/2 are the ends of chord of the x2 + y2 = 16, then its length is ____.
12. If the points (3, 0), (0, 4), (0, 0) and ( k, 4) are concyclic, then k =______.
13. If the lines λx − y + 1 = 0, x − 2y + 3 = 0 cuts coordinate axes in concyclic points, then λ =_____.
14. If the straight lines y = m 1 x + c 1 and y = m 2x + c 2 meets the coordinate axes at concyclic points, then the value of (m1m2 + 4) = _____.
Tangent and Normal
Single Options Correct MCQs
15. A line drawn through the point P(4, 7) cuts the circle x2 + y2 = 9 at the points A and B. Then PAPB is equal to (1) 74 (2) 53 (3) 56 (4) 65
16. If the line 3x – 4y = λ cuts the circle x2 + y2 –4x– 8y– 5 = 0 in two points, then limits of λ are
(1) [-35, 15] (2) (-35, 15)
(3) (-35, 10) (4) (-35, 15]
17. The sum of the minimum and maximum distances of the point (4, -3) to the circle x2 + y2 + 4x– 10y– 7 = 0 is (1) 10 (2) 12 (3) 16 (4) 20
18. The nearest point on the circle x2 + y2 – 6x + 4y – 12 = 0 from the point (–5, 4) is (1) (1, 1) (2) (–1, 1) (3) (–1, 2) (4) (–2, 2)
19. The least distance of the line 8 x – 4y + 73 = 0 from the circle 16x2 + 16 y2 + 48x – 8y – 43 = 0 is____
(1) 5 2 (2) 25
(3) 35 (4) 45
20. If the line y = mx , m ∈ I , lies out side the circle, x2 + y2 − 20y + 90 = 0, then number of integral values of m is ____
(1) 5 (2) 7 (3) 9 (4) 4
21. Consider the circle x2 + y2 – 6x + 4y = 12. The equation of tangent to this circle that is parallel to the line 4 x + 3y +5 = 0 is
(1) 4x + 3y + 10 = 0 (2) 4x + 3y – 9 = 0
(3) 4x + 3y + 9 = 0 (4) 4x + 3y – 31 = 0
22. The equation of the circle with centre at (4, 3) and touching the line 5x – 12y–10 = 0 is
(1) x2 + y2 – 4x– 6y+ 4 = 0
(2) x2 + y2 + 6x– 8y+ 16 = 0
(3) x2 + y2 – 8x– 6y+ 21 = 0
(4) x2 + y2 – 24x– 10y+ 144 = 0
23. The circle passing through (1, –2) and touching the axis of x at (3, 0) also passes through the point
(1) (2, –5) (2) (5, –2)
(3) (–2, 5) (4) (–5, 2)
24. Find the equation of circle having normals
(x − 1)(y − 2) = 0 and a tangent 3x + 4y = 6.
(1) (x − 1)2 + (y − 2)2 = 1
(2) (x − 2)2 + (y − 1)2 = 1
(3) (x + 1)2 + (y + 2)2 = 1
(4) (x + 2)2 + (y + 1)2 = 1
Numerical Value Questions
25. The number of points having integral coordinates lies inside x2 + y2 = 25 is n. Then integral part of 9 n =_____.
26. If the circle x2 + y2 – 6x – 8y +(25 – a2) = 0 touches the axis of x , then a2 equals to______.
27. If the line 3x – 4y – k = 0, (k > 0) touches the circle x2 + y2 − 4x − 8y − 5 = 0 at (a, b), then k + a + b is equal to _____.
28. An infinite number of tangents can be drawn from (1, 2) to the circle x2 + y2 – 2x – 4y + λ = 0, then λ is______.
29. The x-intercept of the circle x2 + y2 + 8x – 9 = 0 is ______.
Pair of Tangents
Single Option Correct MCQs
30. The length of tangent from any point on the circle x2 + y2 + 4x – 6y – 12 = 0 to the circle x2 + y2 + 4x – 6y + 4 = 0 is
(1) 2 (2) 16
(3) 8 (4) 4
31. Slopes of tangents through (7, 1) to the circle x2 + y2 = 25 satisfy the equation
(1) 12m2 + 7m + 12 = 0
(2) 12m2 – 7m + 12 = 0
(3) 12m2 + 7m – 12 = 0
(4) 12m2 – 7m – 12 = 0
32. From any point on the circle x2 + y2 + 2gx + 2fy+c = 0 tangents are drawn to the circle x2 + y2 + 2gx + 2fy+csin2a +(g2 +f2)cos2a = 0, then angle between tangents is______. (1) a (2) 2 a (3) 2 π (4) 0°
33. Tangents are drawn to the circle x2 + y2 = 9 at the points where it is cut by the line 4 x + 3 y = 9 then point of intersection of tangents is
(1) (3, 4) (2) (4, 3)
(3) (–3, 4) (4) (4, –3)
34. From any point on the circle x 2 + y 2 = a 2 tangents are drawn to the circle x 2 + y 2 = a2 sin2θ. The angle between them is (1) θ/2 (2) θ (3) 2θ (4) 4θ
35. Tangents to x2 + y2 = a2 having inclinations a and b intersect at P. If cot a + cot b = 0, then the locus of P is
(1) x + y = 0 (2) x – y = 0 (3) xy = 0 (4) xy = a2
36. Locus of the point of intersection of tangents to the circle x2 + y2 + 2x + 4y – 1 = 0 which include an angle of 60° is
(1) x2 + y2 + 2x + 4y – 19 = 0
(2) x2 + y2 + 2x + 4y + 19 = 0
(3) x2 + y2 – 2x – 4y – 19 = 0
(4) x2 + y2 – 2x – 4y + 19 = 0
37. Locus of the points of intersection of perpendicular tangents drawn one to each of the circles x2 + y2 – 4x + 6y – 37 = 0, x2 + y2 – 4x + 6y – 20 = 0
(1) x2 + y2 – 4x + 6y = 0
(2) x2 + y2 – 4x + 6y – 50 = 0
(3) x2 + y2 – 4x + 6y – 57 = 0
(4) x2 + y2 – 4x + 6y – 70 = 0
38. From the point (3, 4) chords are drawn to the circle x2 + y2 – 4x = 0. Then the locus of mid-point of chords is
(1) x2 + y2 – 5x – 4y + 6 = 0
(2) x2 + y2 + 5x – 4y + 6 = 0
(3) x2 + y2 – 5x + 4y + 6 = 0
(4) x2 + y2 – 5x – 4y – 6 = 0
Numerical Value Questions
39. Tangents are drawn from the point (4, 3) to the circle x2 + y2 = 9. Then the area of the triangle formed by these tangents and the chord of contact is 64 25 ∆ square units where Δ is equal to_____.
40. If the radius of the circumcircle of the triangle TPQ, where PQ is chord of contact corresponding to point T with respect to circle x2 + y2 – 2x + 4y – 11 = 0, is 12 units, then minimum distance of T from the director circle of the given circle is_____.
Relative Position of Two Circles
Single Option Correct MCQs
41. The common tangent at the point of contact of the two circles x 2 + y 2 − 4 x − 4 y = 0, x2 + y2 + 2x + 2y = 0
(1) x + y = 0 (2) x – y = 0 (3) 2x – 3y = 0 (4) x – 2y = 0
42. The set of all real values of λ for which exactly two common tangents can be drawn to the circles x2 + y2 – 4x – 4y + 6 = 0 and x2 + y2 – 10x – 10y + λ = 0 in the interval.
43. The set of values of ‘ λ' for which the real circles x2+y2=λ2 and 22 22220xyxy λ +−−−= have exactly one common tangent is (1) {0, 5} (2) {–3, 5} (3) {-3, 0, 5} (4) {–3}
44. The condition that two circles x2 + y2 + 2ax + c = 0, x2 + y2 + 2by + c = 0 may touch each other is (1) 22 111 abc += (2) 222 111 abc += (3) 222 112 abc += (4) 112 abc +=
45. For the circles x 2 + y 2 + 2 λx + c = 0, x2 + y2 + 2my − c = 0 the number of common tangents when c ≠ 0 is (1) one (2) two (3) four (4) zero
46. If ()2 22 1 :322Cxy+=+ is a circle and PA and PB are a pair of tangents on C1 , where P is any point on the director circle of C1 , then the radius of the smallest circle which touches C 1 externally and also the two tangents PA and PB is
(1) 233 (2) 221
(3) 221 + (4) 1
Numerical Value Questions
47. The number of common tangents to x2 + y2 = 4, (x –3)2 + (y – 4)2 = 9 is______.
48. The circles x2 + y2 = a2 , x2 + y2 – 6x – 8y + 9 = 0 touch externally. Then a = _______.
49. If two circles x2 + y2 + 4x + 6y = 0 and x2 + y2 + 2gx + 2fy = 0 touch each other, then the value of f/g = ______.
50. If the circles x2 + y2 + ax + by + c = 0 and x2 + y2 + bx + ay + c = 0 touch each other then (a + b)2 = λc, where λ is _____.
System of Circles
Single Option Correct MCQs
51. If the angle between the circles x2 + y2 − 4x − 6y − 3 = 0, x2 + y2 + 8x − 4y + λ = 0 is 60°, then the value of λ is ______. (1) –11 (2) 29 (3) 18 (4) –29
52. The equation of the circle passing through the point of intersection of the circles x2 + y 2 − 4 x − 2 y = 8 and x 2 + y 2 − 2 x − 4 y = 8 and the point (-1,4) is
(1) x2 + y2 + 4x + 4y − 8 = 0
(2) x2 + y2 – 3x + 4y + 8 = 0
(3) x2 + y2 + x + y − 8 = 0
(4) x2 + y2 – 3x – 3y − 8 = 0
53. The equation of the circle which passes through the point (3, 2) bisects the circumference of the circle x 2 + y 2 = 15 and cuts the circle x2 + y2 + 4x + 6y + 3 = 0 orthogonally is
(1) x2 + y2 + 6x + 8y – 43 = 0
(2) x2 + y2 + 6x – 8y – 15 = 0
(3) x2 + y2 – 6x + 8y – 11 = 0
(4) x2 + y2 – 6x – 8y + 21 = 0
54. If q is the angle of intersection of two circles x2 + y2 = a2 and (x – c)2 + y2 = b2, then the length of common chord of two circles is
(1)
22 2cos ab abab+−θ
(2) 22 2 2cos ab abab+−θ
(3) 22 2sin 2cos ab abab θ +−θ
(4) 22 sin 2cos ab abab θ +−θ
55. The length of the common chord of the two circles (x – a)2 + (y – b)2 = c2, (x – b)2 + (y – a)2 = c2 is (1) ()2 2 42-cab + (2) ()2 2 42
56. A circle S cuts three circles x2 + y2 − 4x − 2y + 4 = 0, x2 + y2 − 2x − 4y + 1 = 0 and x2 + y2 + 4x + 2y + 1 = 0 orthogonally. Then the radius of S is
(1) 29 8 (2) 28 11 (3) 29 7 (4) 29 5
57. If the circle x2 + y2 + 2x + 3y + 1 = 0 cuts another circle x2 + y2 + 4x + 3y + 2 = 0 in A and B, then the equation of the circle with AB as a diameter is
(1) 2x2 + 2y2 + 2x + 6y + 1 = 0
(2) x2 + y2 + x + 3y + 3 = 0
(3) x2 + y2 + x + 6y + 1 = 0
(4) 2x2 + 2y2 + x + 3y + 1 = 0
58. The equation of the circle which pass through the origin and cuts orthogonally each of the circles x2 + y2 – 6x + 8 = 0 and x2 + y2 – 2x – 2y = 7 is
(1) 3x2 + 3y2 – 8x – 13y = 0
(2) 3x2 + 3y2 – 8x + 29y = 0
(3) 3x2 + 3y2 + 8x + 29y = 0
(4) 3x2 + 3y2 – 8x – 29y = 0
59. A line l meets the circle x2+ y2 = 61 at A, B and P(–5, 6) is such that PA = PB = 10. Then the equation of l is
(1) 5x + 6y + 11 = 0 (2) 5x – 6y – 11 = 0
(3) 5x – 6y + 11 = 0 (4) 5x – 6y + 12 = 0
60. The point from which the tangents to the circles x2 + y2 − 8x + 40 = 0, 5x2 + 5y2 − 25x + 80 = 0 and x2 + y2 − 8x + 16y + 160 = 0 are equal in length is
(1) 15 8, 2
(3) 15 8, 2
(2) 15 8, 2
(4) 15 8, 2
61. If the radical centre of three circles x 2 + y2 − 2x − 1= 0, x2 + y2 − 3y = 1, and 2x2 + 2y2 − x − 7y − 2 = 0 is Q, then Qx +Qy = (1) 3 (2) 0 (3) 1 (4) –1
62. ABC is a triangle. The radical centre of the circles with AB, BC, CA as the diameters is (–6, 5). If A(3, 2), B(2, 1) then C = (1) (1, 1) (2) (1, 2) (3) (2, 3) (4) (1, –2)
Numerical Value Questions
63. The distance of the point (1, 2) from the common chord of the circles x 2 + y 2 – 2 x + 3 y – 5 = 0 and x 2 + y 2 + 10 x + 8 y = 1 is______.
64. If the equation of a circle which passes through the point (1,2) and the points of intersection of the circles x 2 + y 2 – 8 x – 6 y + 21 = 0 and x 2 + y 2 – 2 x – 15 = 0 is x 2 + y 2 + ax + by + 9 = 0, then a2 +b2 is ______.
65. If the length of the common chord of the circles ( x – 2) 2 + ( y – 6) 2 = 9, (x – 4) 2 + (y – 8)2 = 9 is k, then k2 = _____.
Multiple Concept Questions
Single Option Correct MCQs
66. Locus of the centroid of the triangle whose vertices are (acos t, asin t), (bsin t, –bcos t) and (1, 0) where t is a parameter is (1) (3x + 1)2 + (3y)2 = a2 – b2
1. If an equilateral triangle is inscribed in the circle x2 + y2 – 6x – 4y + 5 = 0 then length of its side is
(1) 6 (2) 26 (3) 36 (4) 46
2. A circle is inscribed in an equilateral triangle and a square is inscribed in the circle. The ratio of the area of the triangle to the area of the square is
(1) 3:2 (2) 3:1
(3) 33:2 (4) 3:2
3. A right angled isosceles triangle is inscribed in the circle x2 + y2 – 4x – 2y – 4 = 0 then length of the side of the triangle is (1) 2 (2) 22
(3) 32 (4) 42
4. Two vertices of an equilateral triangle are (-1,0) and (1,0) and its third vertex lies above the X-axis. The equation of the circum circle of the triangle is
(1) x2 + y2 = 1
(2) () 22 3230 xyy++−=
(3) () 22 3230 xyy+−−=
(4) () 22 3230 xyy−−−=
5. The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is 60°. If the area of the quadrilateral is 43, then the perimeter of the quadrilateral is (1) 12.5 (2) 13.2 (3) 12 (4) 13
6. If a line segment AM = a moves in the plane XOY remaining parallel to OX so that left end point A slides along the circle x2 + y2 = a2, then the locus of M is
(1) x2 + y2 = 4a2
(2) x2 + y2 = 2ax
(3) x2 + y2 = 2ay
(4) x2 + y2 – 2ax–2ay= 0
7. The equation of circumcircle of a regular hexagon whose two consecutive vertices have the coordinates (–1, 0) and (1, 0) which lies wholly above x–axis is (1) 22 2310xyy+−−=
(2) 22 310xyy+−−=
(3) 22 2310xyx+−−=
(4) none of these
Numerical Value Questions
8. One of the diameter of the circle circumscribing the rectangle ABCD is 4y = x + 7. If A and B are the points (−3, 4) and (5, 4) respectively, then the area of the rectangle is_____.
9. In Δ ABC the equation of the side BC is 2x− y = 3 and its circum centre and ortho centre are (2,4) and (1,2) respectively then square of radius of the circum circle of Δ ABC is r 2 then [ r 2 ] is_________ (where[.] represents GIF)
10. A circle C 1 is inscribed in a square S 1 of side a 1 . Another square S 2 of side a 2 is incribed C1. A Circle C2 is inscribed in S1.
The process is continued.... If a1 = 42 , and s = a 1 + a 2 +.....∞, then () 21 s =
Tangent and Normal
Single Options Correct MCQs
11. A circle touches the y-axis at the point (0,4) and passes through the point (2, 0). Which of the following lines is not a tangent to this circle?
(1) 4x – 3y + 17 = 0
(2) 3x – 4y – 24 = 0
(3) 3x + 4y – 6 = 0
(4) 4x + 3y – 8 = 0
12. The straight line x + 2 y =1meets the coordinate axes at A and B. A circle is drawn through A, B and the origin. Then the sum of perpendicular distances from A and B on the tangent circle at the origin is
(1) 5 2 (2) 2 5 (3) 5 4 (4) 4 5
13. A variable circle passes through the fixed point A(p,q) and touches x-axis. The locus of the other end of the diameter through A is___
(1) (x − p)2 = 4qy (2) (x − q)2 = 4py
(3) (y − p)2 = 4qx (4) (y − q)2 = 4px
14. A light ray gets reflected from the line x =−2. If the reflected ray touches the circle, x2 + y2 = 4 and point of incident is (−2,−4), then equation of incident ray
(1) 3x + 4y + 22 = 0
(2) 4x + 3y + 28 = 0
(3) 2x + 4y +20 = 0
(4) x + y + 6 = 0
Numerical Value Questions
15. Number of circles which touch both the axes and whose centre lies on x – 2 y = 3 is_______.
16. The value of a in[0, p ] so that () 22 2sincos10xyxαα +++−= having intercept on x–axis always greater than 2 is , k
, then k is ______.
17. A triangle has two of its sides along the axes, its third side touches the circle x2 + y2 – 2ax – 2ay + a2 = 0 If the locus of the circumcentre of the triangle passes through the point (38, –37), then a2 – 2a is equal to K.Then the digit in ten's place is____
18. From a point A on a circle x2 + y2 = 3 chords AB and AC, making equal angles with the normal at A , are drawn. The maximum perimeter of the∆ABC=_____
Pair of Tangents
Single Option Correct MCQs
19. A tangent to the circle x2 + y2 = 1through the point (0, 5) cuts the circle x2 + y2 = 4 at A and B. The tangents to the circle x2 + y2 = 4 at A and B meet at C. The coordinates of C are
(1) 864 , 55
(3) 864 , 55
(2) 864 , 55
(4) (8, 4)
20. If m 1 , m 2 are the slopes of the tangents drawn from a point (1, -3) to the circle x2 + y2 – 6x + 4y+ 12 = 0, then () 22 12 9 mm+= (1) 16 (2) 25 (3) 4 (4) 1
21. P is a point outside a circle. If the farthest distance of P from the circle is 4 times to the shortest distance of P from the circle. Then the angle between the tangents at P is
22. The straight line x – 2y + 5 = 0 intersects the circle x2 + y2 = 25 in points P and Q, the coordinates of the point of intersection of tangents drawn at P and Q to the circle is____.
23. Tangents are drawn to a unit circle with centre at the origin from each point on the line 2x + y = 4.Then the equation to the locus of the middle point of the chord of contact is
(1) 2(x2 + y2) = x + y
(2) 2(x2 + y2)= x + 2y (3) 4(x2 + y2) = 2x + y (4) none
Numerical Value Questions
24. The equation of pair of tangents from origin to a circle is 24xy +7y2 = 0. If the radius of the circle is 3 then the length of the tangent drawn from the origin is _____.
25. If angle between tangents drawn to x 2 + y2 –12x – 16y = 0 at points where curve is cut by 5y = 5x + c(c > 0) is 2 π and c = 10k, then k = _____.
26. The chord of contact of (acosq, asinq) with respect to x2 + y2 = b2 touches x2 + y2 = c2 where a,b,c are roots of x3 – 14x2 + 56x + d = 0 then a is____.
27. If the locus of the mid-point of the line segment from the point (3, 2) to a point on the circle, x2 + y2 = 1 is a circle of radius r, then the length of diameter =____.
28. The area of the quadrilateral formed by the tangents from the point (4, 5) to the circle x 2 + y 2 – 4 x – 2 y + 11 = 0 with a pair of radii joining the points of contact of these tangents is____.
29. The sum of all possible integral values of a such that the angle between the pair of tangents drawn from M(a, a) to the circle x2 + y2 – 2x – 2y – 6 = 0 lies in the range
is equal to ___
3
30. A circle 22 4220xyxyc ++−+= is the director circle of circle S 1 and S 2 is the director circle of circle S1 and so on. If the sum of radii of all these circles is 2, then the value of c is ______.
Relative Position of Two Circles
Single Option Correct MCQs
31. If P is the point of contact of the circles x2 + y2 + 4x + 4y −10 = 0 and x2 + y2 − 6x − 6y + 10 = 0 and Q is their external centre of similitude, then the equation of the circle with P and Q as the extremities of its diameter is
(1) x2 + y2 + 14x + 14y − 26 = 0
(2) x2 + y2 + 5x + 5y − 8 = 0
(3) x2 + y2 + 5x + 5y + 8 = 0
(4) x2 + y2 – 14x – 14y + 26 = 0
32. The range of values of λ for which the circles x2 + y2 = 4 and x2 + y2 − 4λx + 9 = 0 have two common tangents is
(1) 1313 , 88
(2) 13 8 λ <
(3) 13 8 λ > (4) 15 8 λ >
33. If (2, 6) is a centre of similitude for the circle x2 + y2 = 4 and x2 + y2 − 2x − 6y + 9 = 0, the length of the common tangent of circles through it is
(1) 9 (2) 3
(3) 6 (4) 4
34. The area of triangle formed by the common tangents of the circles x2 + y2 + 12x = 0 and x2 + y2 − 4x = 0 is
(1) 83 (2) 123
(3) 8 (4) 16
35. The equation of the circle which touches the circle x2 + y2 − 6x + 6y +17= 0 externally and having the lines x2 − 3xy – 3x + 9y = 0 as two normals, is
(1) x2 + y2 − 2x + 5y − 1 = 0
(2) x2 + y2 + 2x + 3y + 1 = 0
(3) x2 + y2 − 6x – 2y + 1 = 0
(4) x2 + y2 + 4x – 3y + 3 = 0
Numerical
Value Questions
36. If C1: x2 + y2 – 20x + 64 = 0 and C2 : x2 + y2 + 30x + 144 = 0, then the length of the
shortest line segment PQ which touches C1 at P and C2 at Q is ______.
37. If P and Q are variable points on C1 : x2 + y 2 = 4 and C 2 : x 2 + y 2 – 8 x – 6 y + 24 = 0 respectively, then maximum value of PQ is equal to______.
System of Circles
Single Option Correct MCQs
38. If the radical axis of the circles x2 + y2 + 2gx + 2fy + c = 0 and 2x2 + 2y2 + 3x + 8y + 2c = 0 touches the circle x2 + y2 + 2x + 2y + 1 = 0, then
39. If the circle x2 + y2 + 2gx + 2fy + c = 0 bisects the circumference of the circle x 2 + y 2 + 2g ' x + 2f'y + c ' = 0, then the length of the common chord of the circles is (1) 22 2 gfc +− (2) 22 111 2 gfc +− (3)
40. Suppose ax + by + c = 0 where a,b,c are in A.P be a normal to a family of circles. The equation of the circle of the family which intersects the circle x2 + y2 − 4x − 4y − 1 = 0 orthogonally is ‘S' then radius of ‘S' is (1) 8 (2) 7 (3) 6 (4) 8
Numerical Value Questions
41. If the angle between two equal circles with centres(–2, 0), (2, 3) is 120°, then the radius of the circle is _____.
42. The radius of the circle whose centre lies at (1, 2), while cutting the circle x2 + y2 + 4x + 16y – 30 = 0 orthogonally is λ units. Then λ 2 =______.
43. Let S1 ≡ x2 + y2 − 4x − 8y + 4 = 0 and S2 its image in the line y = x. The radius of the circle touching y = x at (1, 1) and orthogonal to S2 is, 3 λ , then λ 2 + 2 = _____.
Multiple Concept Questions
Single Option Correct MCQs
44. The minimum distance from (0, 0) to the locus of the image of (2, 3) with respect to the line (x – 2y + 3) + λ (2x – 3y + 4) = 0 is
(1) 102 (2) 52
(3) 32 (4) 52
45. Two rods of length ‘ a ' and ‘ b ' slide along coordinate axes such that their ends are concyclic. Locus of the centre of the circle is
(1) 4(x2 + y2) = a2 + b2
(2) 4(x2 + y2) = a2 – b2
(3) 4(x2 – y2) = a2 – b2
(4) xy = ab
46. The locus of the centre of a circle of diameter is which rolls on the out side of the circle
x2 + y2 + 3x – 6y – 9 = 0 is
(1) x2 + y2 + 3x – 6y + 5 = 0
(2) x2 + y2 + 3x – 6y – 31 = 0
(3) 22 29 360 4 xyxy++−+=
(4) x2 + y2 + 3x – 6y + 29 = 0
47. A circle of radius 4 units touches the coordinate axes in the first quadrant. The equation of its image with respect to the line mirror y = 0 is
(1) x2 + y2 – 8x + 8y + 16 = 0
(2) x2 + y2 – 8x – 8y + 16 = 0
(3) x2 + y2 – 4x – 4y + 4 = 0
(4) x2 + y2 + 4x – 4y + 4 = 0
48. If y = f ( x ) = ax+b is a tangent to circle
x2 + y2 + 2x + 2y – 2 = 0, then the value of (a+b)2 + 2(a–b)(a+b + 1) is equal to (1) 0 (2) 1 (3) 3 (4) –3
49. From the origin chords are drawn to the circle (x –1)2 + y2 = 1. The equation of the locus of the middle points of these chords is
(1) x2 + y2 – 3y = 0 (2) x2 + y2 – 3x = 0
(3) x2 + y2 – x = 0 (4) x2 + y2 – y = 0
Numerical Value Questions
50. A right angled isoceles triangle is inscribed in the circle x2 + y2 – 6x + 10y – 38 = 0. Then its area is ________ (square units)
51. If the circle 2 + y2 + 8x – 4y + c = 0 touches the circle x2 + y2 + 2x + 4y – 11 = 0 externally and cuts the circle x2 + y2 – 6x + 8y + k = 0 orthogonally, then |k + 3c| = _____.
52. The circles having radii 1, 2, 3 touches each other externally. Then the radius of the circle which cuts the three circles orthogonally is______.
53. Two circles of radii a and b touching each other externally are inscribed in area bounded by 2 1 yx =− and x-axis. If b = 1/2, then 4a =_______.
Level
– III
Single Option Correct MCQs
1. If ()2 22 1 :322Cxy+=+ is a circle and PA and PB are a pair of tangents on C1 , where P is any point on the director circle of C1 , then the radius of the smallest circle which touches C 1 externally and also the two tangents PA and PB is
(1) 233 (2) 221
(3) 221 + (4) 1
2. If r1 and r2 are radii of the smallest and the largest circles, respectively, which pass through (5, 6) and touch the circle (x − 2)2 + y2 = 4 , then r1r2 is
(1) 4 41 (2) 41 4 (3) 5 41 (4) 41 6
3. C1 is a circle of radius 1 touching the x-axis and y-axis. C 2 is another circle of radius greater than 1 and touching the axes as well as the circle C1. Then, the radius of C2 is (1) 322 (2) 322 +
(3) 323 + (4) 1
4. Two circles of radii r1 & r2 (r1 > r2) touch each other externally. Then the radius of the circle which touches both of them externally & also their direct common tangent is
(1) () 12 2 12 rr rr + (2) 12rr
(3) 12 2 rr (4) r1 – r2
5. A circle of radius 2 has center at (2,0) and another circle of radius 1 has center at (5,0). A line is tangent to the two circles at points in the first quadrant. The equation of the tangent is
(1) 1 98 xy+= (2) 1 8 22 xy+=
(3) 1 83 xy+= (4) x + 3 y = 9
6. If the distance from origin to centres of three circles x2 + y2 -2 λ i x = c2 (i = 1,2,3) are in GP then lengths of tangents drawn to them from any point on the circle x2 + y2 = c2 are in (1) AP (2) GP (3) HP (4) AGP
7. The equation of the circle which is touched by y = x has its centre on the positive direction of the x–axis and cuts off a chord of length 2 unit along the line 30 yx−= is
(1) x2 + y2 – 4x + 2 = 0
(2) x2 + y2 – 4x + 1 = 0
(3) x2 + y2 – 8x + 8 = 0
(4) x2 + y2 – 4y + 2 = 0
8. If the curves(x –1) (y – 2)= 5 and (x – 1)2 + ( y + 2) 2 = r 2 are intersect at four distinct points A, B, C, D and the centroid of ∆ABC lies on the line y = 3x – 4, then the locus of D is ______
(1) y = 3x
(2) x2 + y2 + 3x + 1 = 0
(3) 3y = x+ 1
(4) y = 3x+ 1
9. A whe el of radius 8 units rolls along the diameter of a semicircle of radius 25 units if bumps into this semicircle what is the length
of the portion of the diameter that cannot be touched by the wheel.
(1) 12 (2) 15
(3) 17 (4) 20
10. If a circle of radius R passes through the origin O and intersects the coordinate axes at A and B , then the locus of the foot of perpendicular from O on AB is:
(1) (x2 + y2)2 = 4R2x2y2
(2) (x2 + y2)3 = 4R2x2y2
(3) (x2 + y2)2 = 4Rx2y2
(4) (x2 + y2)(x + y) = R2xy
11. If the point (1, 4) lies inside the circle x2 + y2 – 6x – 10y + p = 0 and the circle does not touch or intersect the coordinate axes, then the set of all possible values of p in the interval:
(1) (0, 25) (2) (25, 39)
(3) (9, 25) (4) (25, 29)
12. The circle lying in the first quadrant whose centre lies on the curve y = 2 x 2 − 27 has tangents as 4x − 3y = 0 and the y-axis, then the diameter of the circle is____.
(1) 6 (2) 7 (3) 8 (4) 9
13. Let AB be the chord of contact of the point (5, –5) with respect to the circle x2 + y2 = 5. Then the locus of orthocentre of triangle PAB (where P is any point on the circle) is ____.
(1) (x – 1)2 + (y + 1)2 = 5
(2) 22 5 (1)(1) 2 xy−++=
(3) (x + 1)2 + (y – 1)2 = 5
(4) 22 5 (1)(1) 2 xy++−=
14. From origin chords are drawn to the circle x2 + y2 – 2px = 0, then locus of midpoints of all such chords is_____.
(1) x2 + y2 – px = 0
(2) x2 + y2 + 2px = 0
(3) x2 + y2 – 2px = 0
(4) Does not exists
Numerical Value Questions
15. In a triangle ABC, equation of BC is xy =0, orthocentre and centroid of the triangle are (5,8) and (3,14/3) respectively. If the diameter of circumcircle of the triangle ABC is 10 λ, then the value of λ = ______.
16. If 4l2−5m2+6l+1=0 and the line lx + my + 1 = 0 touches a fixed circle, then (1) the centre of the circle is at the point (4, 0) (2) the radius of the circle is equal to 5 (3) the circle passes through origin (4) none of the above
17. From a point P outside of a circle with centre at C, tangent segments PA and PB are drawn. If 1/(CA)2 + 1/(PA)2 = 1/16, then the length of the chord AB is ______.
18. Tangents are drawn from P(1, 8) to x2 + y2 − 6x − 4y − 11 = 0 touches at A, B. If R is radius of circum circle of triangle PAB. Then [R] = ___.([.] represents greatest integer function).
19. If the variable line 3x + 4y = α lies between the two circles (x – 1)2 + (y – 1)2 = 1 and (x – 9)2 + (y – 1)2 = 4, without intercepting a chord on either circle, then the sum of all the integral values of α is_ ________.
20. If radii of the smallest and largest circle passing through the point () 3,2 and touching the circle 22 2220xyy+−−= are r1 and r2 respectively, then the arithmetic mean of r1 and r2 is ____.
21. The length of the common internal tangent to two circles is 7 and that of a common external tangent is 11. If the product of radii of two circles is p, then the value of p/2 is ____.
22. There are two circles whose equations are x2 + y2 = 9 and x2 + y2 − 8x−6y + n2 = 0, n∈Z. If the two circles have exactly two common tangents, then the number of possible value of n = ____
23. Two circles having radii r1 and r2 passing through vertex A of a triangle ABC. One of the circle touches the side BC at B and other circle touches the side BC at C. If a = 5 and A = 30°, then 12rr = ____
24. The radical centre of the three circles is at the origin. The equation of the two of the circles are x2 + y2 = 1 and x2 + y2 + 4x + 4y – 1 = 0 If the third circle passes through the points (1, 1) and (–2, 1), and its radius can be expressed in the form of p/q, where p and q are relatively prime positive integers, then the value of (p + q) is______.
25. Let C1 and C2 be two circles intersecting at points P and Q. The tangent line closer to Q touches C1 and C2 at M and N respectively. If PQ = 3 QN = 2 and MN = PN, The value of QM2 is____.
26. The straight line 2 x – 3 y = 1 divides the circular region x2 + y2 ≤ 6 into two parts. If 3531111 2,,,,,,, 4244484 S
then the number of points in S lying inside the smaller part is ______.
27. Let a circle C:(x − h)2 + (y − k)2 = r2 , k > 0 touch the x-axis at (1, 0). If the line x + y = 0 intersects the circle ‘C' at ‘P' and ‘Q' such that the length of the chord PQ is ‘2', then the value of h + k + r = _____
28. If two circles touching both the axes intersect at two points P and Q where P = (3,1), then (PQ)2 =_____.
29. Let r1 and r2 be the radii of the largest and smallest circles, respectively, which pass through the point (−4, 1) and having their centers on the circumference of the circle x2 + y2 + 2x + 4y − 4 = 0.If 1 2 2 rab r =+ then a + b is equal to______.
30. A circle C 1 with radius 5 touches x-axis and another circle C2 with radius 4 touches y-axis. These two circles touch each other externally at P (α, β )( α , β, ≥ 0). let the locus of the point P be the curve S. Two perpendicular tangents are drawn to the curve S from Q(a, b). Then the value of (a − 4)2 + (b − 5)2 = ______.
31. Tangents are drawn from an external point P(6, 8) to the circle x2 + y2 = r2. The radius r of circle such that area of triangle formed by the tangents and chord of contact is maximum is ________.
32. Normal at B to the circle x2 + y2 = 5 intersects it again at C. If a line through C intersecting the circle at D meet the tangent to it at B at A(3, 1) then length of CD is ______.
33. If circle x 2 + y 2 = c with radius 3 and
THEORY-BASED QUESTIONS
Very Short Answer Questions
1. If g2 + f2 – c = 0 then the equation x2 + y2 + 2gx + 2fy + c = 0 represents a point, then what are the coordinates of that point?
2. The number of circles passing through the given two points A, B is infinite, among them what is the radius of smallest circle?
3. What is the number of regions that divide a circle in a plane?
4. If A, B, C, D are concyclic points, then what is equal to OA. OB?
5. If the lines a1x + b1y + c1 = 0 and a2x + b2y + c 2 = 0 intersect the coordinate axes at four points then what is the equation of circle passing through those four points ?
6. A square inscribed in a circle x 2 + y 2 + 2gx + 2fy + c = 0 whose sides are parallel to the coordinate axes then what are the coordinates of square?
7. If q is an angle subtended by a chord
x 2 + y 2 + ax + by+c = 0 with 6 radius intersect at two points A and B . If length of ABk = . Then k is _____.
34. AB is any chord of the circle x2 + y2 − 6x − 8y − 11 = 0 which subtends an angle at 2 π (1, 2) If locus of midpoint of AB is a circle x2 + y2 −2ax − 2by −c = 0 ; then the value of (a + b + c) is ______.
35. The sum of abscissa and ordinate of a point on the circle x2 + y2 − 4x + 2y −20=0 which is nearest to 3 2, 2
is ______.
36. Two congruent circles with centres at (2, 3) and (5, 6) intersects at right angle; Then the radius of the circle is _____.
having length p in a circle of radius r, then find the expression for tan q.
8. If OA, OB are two equal perpendicular chords in a circle having centre at ( a, b), then the slopes of OA, OB are the roots of the quadratic equation m2 – 2km – 1 = 0, then find the expression for k in terms of a, b.
9. What is the condition that the circle x2 + y2 + 2gx + 2fy + c = 0 touches y–axis ?
10. What is the centre of the circle inscribed in a square formed by the lines x2 – (a + b)x + ab = 0 and y2 – (c + d)y + cd = 0?
11. Find the equation of circle having centre at (a, b) and touching x–axis.
12. What is the equation of circle touching the line L = 0 at the point P( x1, y1) on it?
13. What is the equation of circle lies in the third quadrant and touching both coordinate axes ?
14. What is the number of circles having radius r touches both coordinate axes?
15. What is the condition that the line xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0 is tangent to the circle S = 0?
Statement Type Questions
Each question has two statements, statement I (S-I) and statement II(S-II). Mark the correct answer as
(1) if both statement I and statement II are correct.
(2) if both statement I and statement II are incorrect.
(3) if statement I is correct but statement II is incorrect.
(4) if statement I is incorrect but statement II is correct.
16. S-I : The radius of the circle 3 x2 + 3y2 + kxy 9x + (k – 6)y + 3 = 0 is 3 2 units.
S-II : The centre of the circle 3 x 2 + 3 y 2 + kxy + 9 x + ( k – 6) y + 3 = 0 is 3 ,1 2
17. S-I : If a point P(x1, y1) lies inside the circle S = 0, then the power of the point is negative
S-II : The shortest distance of any point P with respect to the circle having centre at C and radius r is |CP – r|
18. S-I : If m 1 , m 2 are slopes of tangents drawn from P(x1, y1) to the circle
x2 + y2 = a2 then 11 12 22 1 2 xy mm xa += +
S-II : The slope form of tangent to the circle x2 + y2 = a2 is 2 1 ymxam =±+
19. S-I : The area of the triangle formed by the tangents drawn from point A to the circle S = 0, having centre at C, radius r and its chord of contact
S 1 = 0 is 22prd , where p,d is the perpendicular distance from A, C to S1 = 0
S-II : The area of the triangle is () () 1 baseHeight 2
20. S-I : The image of a circle S = 0 having centre at C and radius r with respect to the line L = 0 is a circle having radius r.
S-II : The image of a circle S = 0 having centre at C and radius r with respect to the line L = 0 is a circle having centre at C.
21. S-I : The circumcircle of the triangle formed by the line 3x + 4y = 12 with coordinate axes is x2 +y2 – 4x – 3y = 0.
S-II : The in circle of the triangle formed by the line 3x + 4y = 12 with coordinate axes is x2 +y2 – 2x – 2y+ 1 = 0.
22. If the tangents drawn from a point P to the circle S = 0 are perpendicular, then the locus of the point P is also another circle S' = 0.
S-I : Two circles S = 0, S' = 0 are concentric.
S-II : The radius of circle S' = 0 is 2 times the radius of circle S = 0.
23. S-I : The number of circles which touches both axes and having same radius is 4.
S-II : The number of circles with same radius and whose centre lies on the line y = x, touches both axes is 2.
24. S-I : A regular n sided polygon inscribed in a circle of radius r then the length of the side of the polygon is 2sin r n π
S-II : A regular polygon of 9 sides having each side of length 2 units inscribed in a circle of radius csc 9 π
25. S-I : The radical axis is the locus of centre of orthogonal circles to the given two circles.
S-II : The radical axis is the locus of a point whose powers with respect to the given two circles is same.
Assertion and Reason Questions
In each of the following questions, a statement of Assertion (A) is given, followed by a corresponding statement of Reason (R) just below it. Mark the correct answer as
(1) if both (A) and (R) are true and (R) is the correct explanation of (A)
(2) if both (A) and (R) are true but (R) is not the correct explanation of (A)
(3) if (A) is true but (R) is false
(4) if (A) and (R) are false
26. (A) : The centre of the circle λx2 +(2λ – 3) y2 – 4x + 6y – 1 = 0 is (2, –3).
(R) : Equation ax2 + by2 + 2gx + 2fy + c = 0 represents circle if and only if a = b, then the centre of the circle is
27. (A) : The equation of smallest circles passing through the points two points (1, 2), (–3, –1) is x2 + y2 +2x – y – 5 = 0 .
(R) : Equation of any circle having A(x1, y 1), B( x 2, y 2)as ends of diameter is (x – x1)(x – x2) + (y – y1)(y – y2) = 0.
28. (A) : The centre of the circle having the portion of line 1 xy ab += intercepted between the coordinate axes is , 22 ab
(R) : The equation of circle passing through (0, 0),( a , 0), (0, b ) is x 2 + y2 – ax – by = 0.
29. (A) : If the chord y = mx+ 1 makes an angle 45° at any point of the major arc of circle x2 + y2 = 1, then the value of m is 2.
(R) : Any chord subtends an angle 2q at the centre of the circle if it makes an angle 3 q at any point on the major arc.
30. (A) : Let Q be any point on the circle having centre at C and radius r, and P is any point in the same plane, then
the range of PQ is [|CP – r|, CP + r]
(R) : The maximum and minimum distances from a point P to the circle having centre at C and radius r are CP + r, |CP – r|.
31. (A) : If two parallel chords makes equal angles 2 q each at the centre of the circle of radius r units, then the distance between those two parallel chords is 2r cos q .
(R) : If two parallel chords make equal angles at the centre then those two chords are equal in length.
32. (A) : If three circles having centres at (x1, y1), (x2, y2) , (x3, y3) and having x–axis as tangent, then 123 111 xxx =+ , x1 < x2 < x3 .
(R) : If a circle touches the x –axis then its radius is equal to absolute value of abscissa of the centre.
33. (A) : In a circle if the chord AB subtends 90° at the centre of the circle, then the radius of the circle is 2 AB
(R) : The angle made by the chord at the centre is double the angle made by the same chord at any point on the arc of same side.
34. (A) : A circle passing through the point P(a,b) in the first quadrant touches the two coordinate axes at the points A and B. The point P is above the line AB. The perpendicular distance from P to AB is 4, then the value of ab is 16.
(R) : Equation of any circle lies in the first quadrant and touches both coordinate axes is x2 + y2 – 2cx – 2cy + c2 = 0 where c > 0.
35. (A) : The radius of the circle which touches both the parallel lines ax + by + c1 = 0 and ax + by + c2 = 0 is 21 22 2 cc ab + .
(R) : The centre of the circle which touch both the parallel lines lies on the mid way of parallel lines.
36. (A) : The locus of a point from which the perpendicular tangents can be drawn to the circle x2 + y2 = a2 is x1 2 + y1 2 = 2a2
(R) : If m 1 , m 2 are slopes of tangents drawn from an external point P( x1, y 1 )to the circle x 2 + y 2 = a 2 , then
37. (A) : The number of common tangents to the circles which intersect at two points is 3
(R) : If C 1C 2 > r 1 + r 2, then two circles touch each other externally
JEE ADVANCE LEVEL
Multiple Option Correct MCQs
1. The number of such points (1,3) aa + where a is any integer lying inside the region bounded by the circles x2 + y2 – 2x
– 3 = 0 and x2 + y2 – 2x – 15 = 0 is
(1) greater than 3 (2) less than 1
(3) greater than 1 (4) less than 2
2. Equation of chord of the circle x2 + y2 – 3x –4y – 4 = 0, which passes through the origin such that the origin divides it in the ratio 4 : 1, is
(1) y = 0 (2) 24x + 7y = 0
(3) 7x + 24y = 0 (4) 7x – 24y = 0
3. The number of circles passing through (2,8) and touching the lines 4x – 3y + 24 = 0 and 4x + 3y – 42 = 0 and having x-coordinate of the centre of the circle less than or equal to 8 is _______.
(1) greater than 2 (2) greater than 1
(3) greater than 3 (4) less than 5
4. From any point P on the circle x2 + y2 – 6x – 8y +15 = 0 tangents PA, PB are drawn to the circle x2 + y2 – 6x – 8y + 20 = 0 and the locus of the orthocentre of the ∆PAB
(R) : Internal centre of similitude is collinear with centre of circles and it lies in between those two centres.
39. (A) : If two circles touch each other externally, then the radical axis is transverse common tangent to both the circles.
(R) : Both circles lie in the opposite side of radical axis.
40. (A) : The radical axis of two circles coincides with common chord of two circles when they intersect each other at two different points.
(R) : The equation of radical axis of common chord of two circles S = 0 and S' = 0 is S – S' = 0.
38. (A) : The external centre of similitude is a point which divides C1C2 in the ratio r1 : r2 externally
is a circle then (1) centre is (3, 4) (2) centre is (–3, 2) (3) radius is 5 (4) radius is 10
5. Let x,y be real variables satisfying x2 + y2 + 8x – 10y – 40 = 0 let a2 = max{(x + 2)2 + (y – 3)2} and b2 = min{(x + 2)2 + ( y – 3)2}, then which of the following is/are true?(a, b ∈ R)
(1) a + b = 18 (2) ab2 += (3) ab42 −= (4) ab = 73
6. The centre of the circles passing through the points (0, 0), (1, 0) and touching the circle x2 + y2 = 9 is /are
(1) 31 , 22
(3) 1 ,2 2
(2) 13 , 22
(4) 1 ,2 2
7. The radius of the circle which touch the lines x + y = 2 and x − y = 2 and the circle x2 + y2 = 1 is
(1) 21 (2) 21 +
(3) ()321 (4) ()321 +
8. A circle C touchs the x-axis and the circle x2 + (y – 1)2 = 1 externally, then locus of the centre of the circle is given by
(1) {(x, y): x2 = 4y} ∪ {0, y): y ≤ 0}
(2) {(x, y): y = x2} ∪ {0, y): y ≤ 0}
(3) {(x, y): x2 + (y – 1)2– 4} ∪ {(0, y): y ≤ 0}
(4) {(x, y): x2 + 4y= 0} ∪ {(0, y): y ≤ 0}
9. Let S1 and S2 be two circles of unit radius with centres at C1 (0,0), C2 (1,0) respectively. S3 is a circle of unit radius passing through C1 and C2 with centre lying above x–axis. The y-intercept of common tangent to S1 and S3 which do not intersect S2 is d1 and y–intercept of common tangent to S2 and S3 which do not intersect S1 is d2 . Then
(1) d1 = 2 (2) 2 23 d =+
(3) 1 12 d =+ (4) 2 22 d =+
10. a, b, c ∈ {2, 3, 4, ......30} and g. c. d of (a, c) is 1. Then the equation of circle with (a, c) as centre and b as radius if 111
abc +=
(1) x2 +y2 – 6x– 4y– 23 = 0
(2) x2 +y2 – 8x– 6y– 119 = 0
(3) x2 +y2 – 10x– 8y– 359 = 0
(4) x2 +y2 – 12x– 10y– 839 = 0
11. If the locus of the points, the sum of the squares of whose distances from n fixed points
Ai(xi, yi), i = 1, 2...... n is equal to k2 is the circle
x2 + y2 + 2gx + 2fy + c = 0 then
(1) 1 i gx n =∑
(2) 1 i fy n =∑
(3) c = k2
(4) () 222 1 ii cxyk n =∑+∑−
12. Three circles of same radius r are drawn such that they touches each other externally. The radius of the circle which touches all the three circles is
(1) () 323 3 r + (2) () 23 3 r
(3) () 22 3 r (4) () 22 3 r +
13. Point Q is moving on the circle (x – 4)2 + (y – 8)2 = 20. Then it broke away from it and moving along a tangent to the circle, cuts x–axis at the point (–2, 0). The coordinates
(1) 346 , 55
(2) 244 , 55
(3) (6, 4) (4) (3, 5)
14. If A(-5,0), B(5,0) and P moves such that 2 1 PA PB = and maximum area of 100 3 PAB ∆= sq.units, then
(1) Locus of the point P is a circle
(2) Number of such triangles PAB are 2
(3) Number of such triangles PAB are 4
(4) Locus of the point P is a Parabola
15. If the line x + y = n, n ∈ N is chord of the circle x2 + y2 = 4, then
(1) Sum of the squares of the length of the chords intercepts by the line on the circle is 22
(2) Number of such chords of 4
(3) Number of such chords of 2
(4) None of these
16. 11 , 22 A is a point on the cirlce x2 + y2 = 1 and B is another point of the circle such that of length 2 AB π = units. Then, coordinates of B can be
(1) 11 , 22
(2) 11 , 22
(3) 11 , 22
(4) none of these
17. Let L 1 be a straight line passing through the origin and L 2 be the straight line x + y = 1. If the intercepts made by the circle x2 + y2 –x +3y = 0 on L1 and L2 are equal, then which of the following equations can represent L1
(1) x + y = 0 (2) x – y = 0
(3) x + 7 y = 0 (4) x – 7 y = 0
18. The centres of the circle having radius 5 and touching the line 4x + 3y – 7= 0 at (1, 1) may be.
(1) (5, 4)
(2) (–5, –4) (3) (–3, –2)
(4) (3, 4)
19. The equation of the tangent to the circle x2 + y2 – 2y = 1, which is perpendicular to the normal which makes equal angles with the coordiante axes is
(1) x – y = 1 (2) y – x = 2
(3) y – x = 3 (4) x – y = 2
20. If a circle passes through (0, 0), (4, 0) and touches the circle x2 + y2 = 36, then (1) The centres of two possible circles are ()() 2,5,2,5
(2) The sum of radii of two circles is 8
(3) The angle between the two possible circles is 1 1 cos 9
(4) The length of common chord of the two possible circles is 4
Numerical/Integer Value Questions
21. The number of integral values of k for which the line, 3x + 4y = k intersects the circle, x2 + y2 − 2x − 4y + 4 = 0 at two distinct points is _____.
22. The number of points P(x, y) lying inside or on the circle x 2 + y 2 = 9 and satisfying the equation tan4x + cot4x + 2 = 4 sin2y is _____.
23. Point A lies on the circle (x −20)2+y2 = 4 and point B lies on the circle x2 + y2 = 36. The mid point of AB is M such that all possible positions of M form a region. If area of the region is 2λπ(λ ∈ N), then λ = _____.
24. If the lines y −2 = m 1( x − 5) and y + 4 = m2(x − 3) intersect at right angles at P and locus of P is x2 + y2 + gx + fy + 7 = 0, then |f + g| is _____.
25. Two circles are given as S1 ≡ x2 + y2 + 14x − 6y + 40 = 0 and S2 ≡ x2 + y2 − 2x + 6y + 7 = 0 with their centres as C1 and C2 respectively. If equation of another circle S 3 whose centre C3 lies on the line 3 x + 4y −16 = 0 and touches the circle S 1 externally and also C1C2 + C2C3 + C3C1 is minimum is x2 + y2 + ax + by + c = 0, then the value of (a + b + c) is ____.
26. The length of transverse common tangent to the circles x2 + y2 – 2x – 6y + 9 = 0 and x2 + y2 + 6x – 2y + 1 = 0 is ____.
27. Consider the family of circles x2 + y2 –2x − 2λy − 8 = 0 passing through two fixed points A and B. then the distance between the points A and B is _____.
28. The circle x2 + y2 − 6x − 10y + λ = 0 does not touch or intersect the coordinate axes and the point (1, 4) is inside the circle. Then the number of integral values of λ are_______.
29. Let the circle x2 + (y − 4)2 = 12 intersects the circle (x − 3)2 + y2 = 13 at A and B. A Quadrilateral ACBD is formed by tangents at A and B to both circles. Then the diameter of circumcircle of quadrilateral ACBD is _________.
30. There are two perpendicular lines, one touches to the circle x2 + y2 = r1 2 and other touches to the circle x2 + y2 = r22. If the locus of point of intersection of these tangents is x2 + y2 = 9, then the value of r1 2 + r2 2 = ______.
Passage-based Questions
(Q. 31-33)
A circle C of radius 1 is inscribed in an equilateral triangle PQR. The points of contact of C with the sides PQ, QR, RP are D, E, F, respectively, The line PQ is given by the equation 360 xy+−= and the point D is 333 , 23
Further, it is given that the origin and the centre of C are on the same side of the line PQ.
31. The equation of circle C is
(1) 22 (23)(1)1 xy−+−=
(2) 2 2 1 (23) 1 2 xy −++=
(3) 22 (3)(1)1 xy−++=
(4) 22 (3)(1)1 xy−++=
32. Points E and F are given by (1)
33 ,,3,0 22
31 ,,3,0 22 (3)
3331 ,,, 2222 (4)
3331 ,,, 2222
33. Equations of the sides QR, RP are (1) 22 1, 1 33 yxyx =+=−−
(2) 1 ,0 3 yxy==
(3) 33 1, 1 22 yxyx =+=−−
(4) 3,0yxy==
(Q. 34-35)
Let P is a point on the circle x2 + y2 − 6x − 8y + 16 = 0, where O is the origin and OX is the positive side of the x–axis then the coordinates of the point P such that 34. OP is minimum
(1) 432 , 55
(3) 68 , 55
(2) 86 , 55
(4) 3224 , 55
35. ∠ POX is maximum (1) (0, 4) (2) 2896 , 2525
(3) 2432 , 55
(Q. 36-37)
(4) 86 , 55
A tangent PT is drawn to the circle x 2 + y 2 = 4 at the point ()3,1. PA straight line L , perpendicular to PT is a tangent to the circle (x − 3)2 + y2 = 1
36. A possible equation of L is (1) 31xy−= (2) 31xy+= (3) 31xy−=− (4) 35xy+=
37. A common tangent of the two circles is (1) x = 4 (2) y = 2
(3) 34xy+= (4) 226xy+=
(Q. 38-39)
The circles x2 + y2 + 2x = 0 and x2 + y2 – 6x = 0 touch each other externally. Then
38. The triangle formed by the direct common tangents and transverse common tangent is (1) an isosceles (2) an equilateral (3) right angled Isosceles (4) right angled
39. The points of intersection transverse common tangent with direct common tangents are
(1) () 0,3 ± (2) () 0,2 ±
(3) () 0,3 ± (4) () 0,23 ±
(Q. 40-42) C2 =(–3, 1) C1
List – I
(I) Possible equation of AB is y = 4
(II) Possible equation of CD is y = 2
(III) Locus of the centres of the circle which intersect given circles orthogonally is 2x + y −3 = 0
(IV) Equation of PQ is x − 2y + 5 = 0
List – II
(i) AB = 4
(ii) CD = 2
(iii) Diameter of the smallest circle touching both the circles is 252
(iv) Diameter of the circle just sufficient to contain both the circles is 254 +
List – III
(P) 1 1 2tan 2 α=
(Q) 1 2tan2 β=
(R) Distance of P from origin is 5 2
(S) Distance of Q from origin is 5
40. Which of the following is the only incorrect combination?
41. Which of the following is the only INCORRECT combination?
(1) (I) (ii) (P) (2) (I) (ii) (S) (3) (IV) (ii) (P) (4) (IV) (ii) (Q)
42. Which of the following is the only incorrect combination?
(1) (I) (iv) (P) (2) (I) (iv) (S) (3) (IV) (iv) (Q) (4) (IV) (iv) (S)
(Q. 43-44)
A circle C 1 of radius r untis rolls outside the circle C 2 : x 2 + y 2 + 2 rx = 0 touching it externally. The line of centres has an inclination 60°. Then
43. The point of contact of C1 and C2 is (1) () ,3rr (2) () ,3rr
(3) 3 , 22 rr
(4) 3 , 22 rr
44. The transverse common tangent is (1) 30xyr++= (2) ++=320xyr
(3) +−=30xyr
(4) +−=320xyr
(Q. 45-46)
A tangent PT is drawn to the circle x 2 + y 2 = 4 at the point () 3,1 . A straight line L, perpendicular to PT is a tangent to the circle (x – 3)2 + y2 = 1
45. A line which touches both the circles is (1) x = 4 (2) y = 2
(3) 34xy+= (4) 226xy+=
46. A possible equation of L is (1) 31xy−= (2) 31xy+= (3) 31xy−=− (4) 35xy+=
(Q. 47-49)
Let C = x2 + y2 + 4x = 0 is a given circle and a circle C1 of radius 2 units rolls on the outer side of the circle ‘C' touching it externally. If the line joining the centres of C and C1 makes an angle 60° with the x-axis, then that circle be C2
47. The locus of centre of the circle C1 is
(1) x2 + y2 + 4x – 12 = 0
(2) 22 4380xyy+−+=
(3) x2 + y2 + 4x + 12 = 0
(4) x2 + y2 – 43y – 12 = 0
48. The equation of least circle containing both the circles C and C1 is
(1) x2 + y2 – 4x + 12 = 0
(2) x2 + y2 – 4x – 12 = 0
(3) 22 223120xyxy+−++=
(4) 22 223120xyxy++−−=
49. The equation of circle joining the centres C and C1 as a diameter is
(1) 22 2230xyxy+−+=
(2) 22 2230xyxy+−−=
(3) 22 2230xyxy++−=
(4) None of these
(Q. 50-51)
The director circle of a circle is the locus of points from which tangents to the circle are at right angles. C1, C2, C3 ,............ is a sequence of circles such that C r +1 is the director circle of C r let equation of C1 be x2 + y2 = 4.
50. The length of a chord of C11 which touches C10 is k, then 2 k = ______.
51. The distance of the chord of contact of tangents of a point on C11 with respect to C10 from the centre of C10 is ______.
(Q. 52-53)
Consider the relation 4l2 – 5m2 + 6l + 1 = 0 where l, m∈ R then the line lx + my + 1 = 0 touches a fixed circle
52. Number of tangents which can be drawn from the point (2, –3) are ______
53. From a point P (2, –3) two tangents are drawn to the circle and A, B are points of contact then PA. PB is _____.
(Q. 54-55)
Consider the circles 22 1 :46120Sxyxy+−−+= and ()()() 22 2 2 :561Sxyr−+−=>
54. S1 and S2 touch internally then(r – 1) 2 =
55. S1 and S2 touch externally then r 2+2r+3 =
Matrix Matching Questions
56. A variable straight line through A(−1, 1) is drawn to cut the circle x 2 + y 2 = 1 at the points B and C. A point P is chosen on the line ABC satisfying the condition given the List–I. Let d be the minimum distance of the origin from the locus of P given in the List–II
List – I
List – II
(A) AB, AP, AC are in AP (p) 0
(B) AB, AP, AC are in GP (q) 1/2
(C) AB, AP, AC are in HP (r) 2
(D) AB, , 2 AP AC are in AP (s) 21
(1) A → r; B → q; C → s; D → p
(2) A → q; B → r; C → p; D → s
(3) A → p; B → s; C → q; D → r
(4) A → r; B → q; C → s; D → p
57. Let A(x1, y1), B(x2, y2), C(x3, y3) be 3 distinct points lying on circle S: x2 + y2 = 1, such that x1x2+ y1y2 + x2x3 + y2y3 + x3x1 + y3y1= 3 2 .
List – I
List – II
(A) Let P be any arbitrary point lying on S, then (PA)2+(PB)2 +(PC)2 = (p) 3
(B) Let the ⊥ ar dropped from point A to BC meets S at Q and OBQk ∠=π where ‘O' is origin, then k = (q) 4
(C) Let R be the point lying on line x + y = 2 at the minimum distance from S and the square of maximum distance of R from S is , abb + then a+ b = (r) 5
(D) Let I and G represent incentre and centroid of ∆ABC respectively, then IA + IB + IC + GA + GB + GC = (s) 6
(1) A → q; B → s; C → r; D → s
(2) A → s; B → r; C → q; D → s
(3) A → r; B → s; C → s; D → q
(4) A → s; B → p; C → r; D → s
58. Match the items of List – I with the items of List – II and choose the correct option.
List –I
List – II
(A) The circle x2 + y2+ 2x + c = 0 and x2 + y2 + 2y + c = 0 touch each other (p) if c = 1
(B) The circle x2 + y2 + 2x +3y + c2 = 0 and x2 + y2 – x +2y + c2 = 0 intersect orthogonally (q) if c = –1
(C) The circle x2 + y2 = 9 contains the circle x2 + y2 − 2x + 1 − c2 = 0 (r) if c = 1 2
(D) The circle x2 + y2 = 9 is contained in the circle x2 + y2 − 6x –8y + 25 –C2 = 0 (s) if c > 8
(1) A → r; B → p,q; C → p,q,r; D → s
(2) A → q,p; B → r; C → p,q,r; D → r
(3) A → p,q,r; B → r; C → r; D → p,q
(4) A → r, B → r, C → p,q; D → p,q,r
59. From the point P(4,–4) tangents PA and PB are drawn to the circle x 2 + y 2 – 6 x +2 y + 5 = 0 whose centre is C.Then Match the items of List – I with the items of List – II and choose the correct option.
List –I
List –II
(A) Length of AB (p) 5 2
(B) Tangent of the angle between PA and PC (q) 10
(C) Area of triangle PAB (r) 1
(D) Absolute difference of slopes of PA and PB (s) 3 4
(1) A → q; B → r; C → p; D → p
(2) A → r; B → q; C → p; D → p
(3) A → p; B → r; C → q; D → p
(4) A → q; B → q; C → r; D → p
60. AB is chord of the circle x 2 + y 2 = 25 whose middle point is M. Then match the following loci of M under various conditions.
List –I
List –II
(A) When AB subtends 60º at centre (p) x2 + y2+ 2y = 0
(B) When AB always passes through (0, –2) (q) x2 + y2 = 0
(C) When AB has constant length 8 (r) x2 + y2 = 75 4
(D) When AB subtends 45° on the circumference of the circle (s) 2(x2 + y2) = 25
(1) A → r; B → p; C → q; D → s
(2) A → s; B → q; C → p; D → r
(3) A → q; B → s; C → r; D → p
(4) A → p; B → r; C → s; D → q
61. Each side of a square has length 4 units and its centre is at (3,4). If one of the diagonals is parallel to the line y = x
List –I
List –II
(A) If P, Q are the vertices of a square and they are farthest and nearest points from origin to the square then P Q is (p) ()2221
(B) The radius of the circle inscribed in the triangle formed by any three vertices is (q) ()221
(C) The radius of the circle inscribed in the triangle formed by any two vertices of square and the centre is (r) 42
(D) The radius of circle inscribed in the square is (s) ()221 (t) 2
(1) A → r; B → p; C → s; D → t
(2) A → s; B → r; C → q; D → p
(3) A → t; B → r; C → p; D → q
(4) A → s; B → t; C → r; D → p
62. Let C1 and C2 be two circles whose equations are x2 + y2 – 2x = 0 and x2 + y2 + 2x = 0 and P ( λ , λ ) is a variable point.
List –I
List –II
(A) P lies inside C1 but outside C2 (p) λ∈ (– ∞ , –1) ∪ (0, ∞ )
(B) P lies inside C2 but outside C1 (q) λ∈ (– ∞ , –1) ∪ (1, ∞ )
(C) P lies outside C1 but outside C2 (r) λ∈ (– 1 , 0)
(D) P does not lie inside C2 (s) λ∈ (0, 1)
(1) A → r; B → p; C → s; D → q
(2) A → s; B → r; C → q; D → p
(3) A → s; B → r; C → p; D → q
(4) A → s; B → q; C → r; D → p
63. Let x2 + y2 +2gx + 2fy + c = 0 be an equation of circle
List –I
List –II
(A) If circle lies in first quadrant, then (p) g < 0
(B) If circle lies above x–axis, then (q) g > 0
(C) If circle lies on the left of y–axis, then (r) g2 –c < 0
BRAIN TEASERS
1. If the radius of the circle (x – 1)2 + (y – 2)2 = 1 and (x – 7)2 + (y – 10)2 = 4 are increasing uniformly with respect to times as 0.3 and 0.4 unit/sec, then they will touch each other at t equal to
(1) 45 s (2) 90 s
(2) 11 s (4) 135 s
2. The circles x2 + y2 – 4x – 81 = 0, x2 + y2 + 24 x – 81 = 0 intersect each other at the points A and B. A line through point A meet one circle at P and a parallel line through B meet the other circle at Q. Then the locus of the mid point of PQ is
(1) (x + 5)2 + (y – 5)2 = 25
(2) x2 + y2 + 10x = 0
(3) x2 + y2 – 4x = 0
(4) x2 + y2 + 2x = 0
3. Tangents PA and PB are drawn to the circle x2 + y2 = 4 from an externa; point P lying on the line y = x such that the angle q between the tangents satisfies 2 33 ππ θ ≤≤ . The coordinates of the point P, if the perimeter of triangle PAB is maximum are
(D) If circle touches positive x–axis and does not intersect y–axis, then (s) c >0
(1) A → r; B → p; C → s; D → q
(2) A → p,r,s; B → r,s; C → q,s; D → p,s
(3) A → s; B → r; C → p; D → q
(4) A → s; B → q; C → r; D → p
(1) () 22,22 (2) 2222 , 33
(3) (2, 2) (4) () 2,2
4. Let O be the centre of the circle x2 + y2 = r 2 where 5 2 r > . suppose PQ is a chord of this circle and the eqaution of the line passing through P and Q is 2x + 4y = 5. If the centre of the circum circle of triangle OPQ lies on the line x + 2 y = 9 then the value of r is (1) 3 (2) 2 (3) 5 (4) 9
5. Let C be a circle passing through the points A(2, –1) and B(3, 4). The line segment AB is not a diameter of C. If r is radius of C and its center lie on the circle (x – 5)2 + (y – 1)2 = 13/2, then r2 is equal to (1) 32 (2) 65/2 (3) 61/2 (4) 30
6. Let the tangent to the circle C 1: x 2 + y 2 = 2 at the point M(–1, 1) intersect the circle C 2: ( x – 3) 2 + ( y – 2) 2 = 5 at two distinct points A and B. If the tangents to C2 at the points A and B intersect at N, then the area of the triangle ANB is equal to (1) 1/2 (2) 2/3 (3) 1/6 (4) 5/3
7. A circle C1 passes through the origin O and as diameter 4 on the positive x– axis. The line y = 2x gives a chord OA of a circle C 1. Let C2 be the circle with OA as a diameter. If the tangetnt to C2 at the point A meets the x–axis at P and y– axis at Q, then QA : AP is equal to (1) 1 : 4 (2) 1 : 5 (3) 2 : 5 (4) 1 : 3
8. In a right angled triangle BC = 5, AB = 4, AC = 3. Let S be the circum circle. Let S1 be the circle touching both sides AB and AC and circle S internally. Let S 2 be the circle touching the produced sides AB and AC of triangle ABC and touching the circle S externally. If r1, r2 are radii of circles S1 and S2 respectively then r1r2 is equal to (1) 12 (2) 20 (3) 15 (4) 24
9. Two circles with radii R and r (R > r) touch each other internally at A. Triangle ABC is an equilateral triangle with B on one circle and C on other. Then the length of side of an equilateral triangle ABC is
(1) 22 23 Rr RRrr −+
FLASHBACK (P revious JEE Q uestions )
JEE Main
1. Four distinct points (2k, 3k), (1, 0), (0, 1) and (0, 0) lie on a circle, for k equal to (27th Jan 2024 Shift 1)
(1) 2 13 (2) 3 13 (3) 5 13 (4) 1 13
2. If the circles (x+1)2+(y+2)2 = r2 and x2+y2–4 x–4y+4 = 0 Intersect at exactly two distinct points, then (30th Jan 2024 Shift 1)
(1) 5< r < 9 (2) 0 < r< 7
(3) 3< r < 7 (4) 1 r7 2 <<
10. Consider a triangle ∆ whose two sides lie on the x–axis and the line x + y + 1 = 0 if the orthocenter of triangle ∆ is (1, 1) then the equation of the circle passing through vertices of the triangle ∆ is
(1) x2 + y2 – 3x + y = 0
(2) x2 + y2 + x + 3y = 0
(3) x2 + y2 + 2y– 1 = 0
(4) x2 + y2 + x + y = 0
11. Let four circles having radii r1 = 5 units, r2 = 5, r3 = 8 units and r4 units are mutually touching each other externally then r 4 is equal to
(1) 9/8 (2) 8/7 (3) 7/8 (4) 8/9
12. Given two circles 22 32()0xyxy+++= and 22 52()0xyxy+++= . Let the radius of the third circle which is tangent to the two given circles and to their common diameter be 21 p p The value of 2p, is (1) 8 (2) 5 (3) 16 (4) 10
3. If one of the diameters of the circle x2+y2–10 x+4y+13=0 is a chord of another circle C, whose center is the point of intersection of the lines 2x+3y = 12 and 3x–2y = 5, then the radius of the circle C, is (31st Jan 2024 Shift 1) (1) 20 (2) 4 (3) 6 (4) 32
4. Let C: x2+y2 = 4 and C': x2+y2–4λx+9 = 0 be two circles. If the set of all values of λ so that the circles C and C' intersect at two distinct points, is R-[a, b], then the point (8a + 12,16 b–20) lies on the curve
(1st Feb 2024 Shift 1)
(1) x2+2y2–5x+6y=3 (2) 5x2 –y = –11
(3) x2–4y2 = 7 (4) 6x2+y2 = 42
5. Let the locus of the mid points of the chords of circle x 2 + ( y – 1) 2 = 11 drawn from the rigin intersect the line x + y = 1 at P and Q. Then, the length of PQ , is (1 st Feb 2024 Shift 2)
(1) 1 2 (2) 2
(3) 1 2 (4) 1
6. A circle touches both the y-axis and the line x + y = 0. Then the locus of its centre, is (25 th Jun 2022 Shift 2)
7. Let a circle C touch the lines L1 : 4x –3y + K1 = 0 and L2 : 4x–3y + K2 = 0, K1, K2 ∈ R. If a line passing through the centre of the circle C intersects L 1 at (-1, 2) and L 2 at (3,-6), then the equation of the circle C is (25 th Jun 2022 Shift 1)
(1) (x –1)2 + (y – 2)2 = 4
(2) (x +1)2 + (y – 2)2 = 4
(3) (x –1)2 + (y + 2)2 = 16
(4) (x –1)2 + (y – 2)2 = 16
8. Let C be a circle passing through the points
A(2, –1) and B(3, 4). The line segment AB is not a diameter of C. If r is the radius of C and its centre lies on the circle 22 13 (5)(1) 2 xy−+−= , then r2 is equal to (26 th Jun 2022 Shift 1)
(1) 32 (2) 65 2
(3) 61 2 (4) 30
9. The set of values of k for which the circle C : 4 x 2 + 4 y 2 − 12 x + 8 y + k = 0 lies inside the fourth quadrant and the point 1 1, 3 lies on or inside the circle C is (27 th Jun 2022 Shift 2)
(1) An empty set (2)
(3) 80 ,10 9
95 6, 9
(4) 92 9, 9
10. If the tangents drawn at the points O(0,0) and () 15,2 P + on the circle x 2+ y2 − 2x − 4y = 0 intersect at the point Q, then the area of the triangle OPQ is (28th Jun 2022 Shift 1)
(1) 35 2 + (2) 425 2 + (3) 535 2 + (4) 735 2 +
11. Let the tangent to the circle C1:x2 + y2 = 2 at the point M (–1,1) intersect the circle C 2: ( x − 3) 2 + ( y − 2) 2 = 5 at two distinct points A and B. If the tangents to C2 at the points A and B intersect at N, then the area of the triangle ANB is equal to (1) 1 2 (2) 2 3 (3) 1 6 (4) 5 3
12. Let the locus of the centre ( α , β ), β > 0 , of the circle which touches the circle x2 + (y − 1)2 =1 externally and also touches the x -axis be L. Then the area bounded by L and the line y = 4 is (25th Jul 2022 Shift 1)
(1) 322 3 (2) 402 3 (3) 64 3 (4) 32 3
13. Let the abscissa of the two points P and Q on a circle be the roots of x2 − 4x − 6 = 0 and the ordinates of P and Q be the roots of y2 + 2y − 7 = 0. If PQ is a diameter of the circle x2 + y2 + 2ax + 2by + c = 0, then the value of (a + b – c) is (26th Jul 2022 Shift 2) (1) 12 (2) 13 (3) 14 (4) 16
14. A circle C1 passes through the origin O and has diameter 4 on the positive x-axis. The line y = 2x gives a chord OA of a circle C1. Let C2 be the circle with OA as a diameter. If the tangent to C2 at the point A meets the x-axis at P and y-axis at Q, then QA : AP is equal to (27 th Jul 2022 Shift 2)
(1) 1 : 4 (2) 1 : 5 (3) 2 : 5 (4) 1 : 3
15. If the circle x 2 + y 2 − 2 gx + 6 y − 19 c = 0, g, c ∈ passes through the point (6, 1) and its centre lies on the line x – 2cy = 8, then the length of intercept made by the circle on x-axis is (27th Jul 2022 Shift 1)
(1) 11 (2) 4 (3) 3 (4) 223
16. Let the tangents at two point A and B on the circle x2 + y2 – 4x +3=0 meet at origin O (0,0). Then the area of the triangle of OAB is (28 th Jul 2022 Shift 2)
(1) 33 2 (2) 33 4 (3) 3 23 (4) 3 43
17. If the tangents at the points P and Q on the circle x2 + y2 – 2x + y = 5 meet at the point 9 , 2 4 R , then the area of the triangle PQR is (6th Apr 2023 Shift 2) (1) 13 8 (2) 5 8 (3) 5 4 (4) 13 4
18. Let O be the origin and OP and OQ be the tangents to the circles x2 + y2 – 6x + 4y + 8 = 0 at the points P and Q on it. If the circumcircle of the triangle OPQ passes through the point 1 , 2 α , then the value of α is (8 th Apr 2023 Shift 2) (1) 3 2 (2) 5 2 (3) 1 (4) 1 2
19. Let A be the point (1,2) and B be any point on the curve x 2 + y 2 = 16. If the centre of the locus of the point P, which divides the lines segment AB in the ratio 3 : 2 is the point C ( α , β ), then the length of the line segment AC is (10th Apr 2023 Shift 2)
(1) 25 5 (2) 65 5 (3) 35 5 (4) 45 5
20. Let centre of a circle C be ( α , β ) and its radius r < 8. Let 3x + 4y = 24 and 3x – 4y = 32 be two tangents, and 4 x + 3y = 1 be a normal to C. Then (α − β + r) is equal to (13 th Apr 2023 Shift 2)
(1) 9 (2) 7 (3) 6 (4) 5
21. The number of common tangents to the circles x 2 + y 2 –18 x –15 y + 131 = 0 and x2+ y2 – 6x– 6y – 7 = 0, is
(15 th Apr 2023 Shift 1)
(1) 2 (2) 1 (3) 3 (4) 4
22. The locus of the mid points of the chords of the circle C 1 : ( x − 4) 2 + ( y − 5) 2 = 4, which subtend an angle θ1 at the centre of the circle C1, is a circle of radius r1. If 13 2 , 33 ππθθ== and +−22 122 1rr rr , then θ2 is equal to (24 th Jan 2023 Shift 2)
(1) 4 π (2) 6 π (3) 3 4 π (4) 2 π
23. The points of intersection of the line ax + by = 0, (a≠b) and the circle x2 + y2 − 2x = 0 are A(α, 0) and B(1, β)The imageof the circle with AB as a diameter in the line x + y + 2= 0 is
(25 th Jan 2023 Shift 1)
(1) x2 + y2 + 3x + 3y + 4 = 0
(2) x2 + y2 + 5x + 5y + 12 = 0
(3) x2 + y2 + 3x + 5y + 8 = 0
(4) x2 + y2+ 5x + 5y + 12 = 0
24. Let the tangents at the points A(4, −11) and B(8, −5) on the circle x2 + y2 − 3x + 10y −15 = 0, intersect at the point C. then the radius of the circle, whose centre is C and the line joining A and B is its tangent, is equal to
(29 th Jan 2023 Shift 1)
(1) 13 (2) 33 4
(3) 213 (4) 213 3
25. Let a circle C1 be obtained on rolling the circle x 2 + y 2 − 4 x − 6 y + 11 = 0 upwards 4 units on the tangent T to it at the point (3, 2). Let C 2 be the image of C 1 in T Let A and B be the centres of circles C1 and C2 respectively, and M and N be respectively the feet of perpendiculars drawn from A and B on the x– axis. Then the area of the trapezium AMNB is
(31 st Jan 2023 Shift 1)
(1) 322 + (2) ()222 + (3) ()212 + (4) ()412 +
26. The set of all values of a2 for which the line x + y = 0, bisects two distinct chords drawn from a point 11 , 22 aa P +− on the circle 2x2 + 2y2 − (1 + a)x −(1 − a)y = 0 is equal to (31st Jan 2023 Shift 2) (1) (8, ∞) (2) (0, 4] (3) (4, ∞) (4) (2, 12]
27. Consider a circle (x–a)2+(y–b)2 = 50, where, a,b >0. If the circle touches the line y+x = 0 at the point P, whose distance from the origin is 42 , then( a + b )2 is equal to ___ ___. (27th Jan 2024 Shift 2)
28. Equation of two diameters of a circle are 2x–3 y = 5 and 3x–4y = 7. The line joining the points 22 ,4 7
and 1 ,3 7
intersects the circle at only one point P( a , b ). Then 17 b – a is equal to ______.
(29th Jan 2024 Shift 1)
29. Consider two circles C 1 : x 2 + y 2 = 25 and C2:(x- a )2+y2 = 16, where a ∈ (5,9). Let the angle between the two radii (one to each circle) drawn from one of the intersection points of C1 and C2 be 1 63 sin 8
. If the length of common chord of C1 and C2 is b, then the value of ( ab )2 equals _____.
(30 th Jan 2024 Shift 2)
30. Let the abscissa of the two points P and Q be the roots of 2x2 – rx + p = 0 and the ordinates of P and Q be the roots of y2 – sy –q = 0. If the equation of the circle described on PQ as diameter is 2(x2 + y2) – 11x – 14y – 22 = 0, then the value of 2 r + s – 2q + p = ______. (26 th Jul 2022 Shift 2)
31. Let a circle C of radius 5 lie below the x–axis. The line L1= 4x + 3y −2 passes through the centre P of the circle C and intersects the line L 2 :3 x − 4 y − 11= 0 at Q . The line L2 touches C at the point Q. Then the distance of P from the line 5x − 12y + 51 = 0 is ______. (27 th Jun 2022 Shift 2)
32. A rectangle R with end points of the one of its sides as (1, 2) and (3,6) is inscribed in a circle. If the equation of diameter of the circle is 2x – y + 4 = 0, then the area of R is ______. (th Jun 2022 Shift 2)
33. If one of the diameters of the circle x2 + y2 − 22x − 62y +14 = 0 is a chord of the circle 2 (22)(22)22 xyr −+−= , then the value of r2 is equal to _______.
(28th Jun 2022 Shift 2)
34. Let the lines 21177yx+=+ and 221167 yx+=+ be normals to the circle C:(x − h)2 +(y − k)2 = r2 . If the line 577 11311 3 yx−=+ is a tangent to the circle C then the value of (5 h − 8k)2 + 5r2 is equal to (28 th Jun 2022 Shift 1)
35. If the circles x2 + y2 + 6x + 8y + 16 = 0and ()() 22 233246xyxx++−++−
6386,0ykk =++>
touch internally at the point P(α, β), then 22 (3)(6) αβ+++ is equal to _______.
(25th Jul 2022 Shift 2)
36. Let the mirror image of a circle C 1 : x 2 + y2 – 2x – 6y + α = 0 in line y = x + 1 be C2 : 5x2 + 5y2 + 10gx + 10fy + 38 = 0. If r is the radius of circle c 2, then α+6 r 2 is equal to ______. (29 th Jul 2022 Shift 1)
37. Let the point (p, p+1) lie inside the region () {} 2 ,:39,03Exyxyxx =−≤≤−≤≤ . If the set of all values of p is the interval (a, b), then b2 + b – a2 is equal to _____.
(6 th Apr 2023 Shift 1)
38. A circle passing through the point P( α , β ) in the first quadrant touches the two coordinate axes at the point A and B. The point P is above the line AB. The point Q on the line segment AB is the foot of perpendicular from P on AB. If PQ is equal to 11 units, then the value of αβ is ______.
(6th Apr 2023 Shift 1)
39. Consider a circle C1 : x2 + y2−4x − 2y = α − 5. Let its mirror image in the line y = 2x + 1 be another circle C2 : 5x2 + 5y2 − 10fx − 10gy + 36 = 0. Let r be the radius of C 2. Then α + r is equal to ____.(8th Apr 2023 Shift 1)
40. Two circles in the first quadrant of radii r1 and r2 touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line x + y = 2. Then +−22 122 1rr rr is equal to _______. (12 th Apr 2023 Shift 1)
41. Point P (-3, 2), Q (9, 10) and R ( a , 4) lie on a circle C with PR as its diameter. The tangents to C at the points Q and R intersect at the point S. If S lies on the line 2x – ky = 1 then k is equal to (25 th Jan 2023 Shift 2)
42. A circle with centre (2,3) and radius 4 intersects the line x + y = 3 at the points P and Q. If the tangents at P and Q intersect at the point S( α , β ) , then 4 α −7 β is equal to ______. (29 th Jan 2023 Shift 2)
43. Let P(a1, b1) and Q(a2, b2) be two distinct points on a circle with centre () 2, 3 C . Let O be the origin and OC be perpendicular to both CP and CQ. If the area of the triangle OCP is 35 2 , then 2222 1212 aabb +++ is equal to ______. (30th Jan 2023 Shift 2)
JEE Advanced
44. Let C1 be the circle of radius 1 with enter at the origin. Let C2 be the circle of radius r with center at the point A = (4,1), where 1 < r < 3. Two distinct common tangents PQ and ST of C 1 and C 2 are drawn. The tangent PQ touches C 1 at P and C 2 at Q . The tangent ST touches C1 at S and C2 at T. Midpoints of the line segments PQ and ST are joined to form a line which meets the x-axis at a point B. If 5 AB , then the value of r2 is equal to _______ (2023 P2)
45. Let ABC be the triangle with AB = 1, AC = 3 and ∠ ABC = 2 π . If a circle of radius r > 0 touches the sides AB , AC and also touches internally the circumcircle of the triangle ABC , then the value of r is _______. (2022 P1)
46. Let G be a circle of radius R > 0. Let G1, G2,…, Gn be n circles of equal radius r > 0. Suppose each of the n circles G1, G2,…, Gn touches the circle G externally. Also, for i = 1, 2,…, n −1, the circle G i touchesG i+1 externally, and G n touches G 1 externally. Then, which of the following statements is/are TRUE ? (2022 P2)
(1) If n = 4, then () 21 rR−<
(2) If n = 5, then r < R
(3) If n = 8, then () 21 rR−<
(4) If n = 12, then ()231 rR+>
47. Consider a triangle Δ whose two sides lie on the x-axis and the line x + y + 1 = 0. If the orthocenter of Δ is (1,1), then the equation of the circle passing through the vertices of the triangle Δ is (2021 P1)
(1) x2+y2−3x + y = 0
(2) x2+y2+ x + 3y = 0
(3) x2+ y2 +2y – 1 = 0
(4) x2+y2+x + y = 0
48. Let O be the centre of the circle x 2 + y 2 = r2 , where 5 2 r > . Suppose PQ is a chord of this circle and the equatio n of the line passing through P and Q is 2x + 4y = 5 . If the centre of the circumcircle of the triangle OPQ lies on the line x + 2y = 4 , then the value of r is ______.(2020 P2)
CHAPTER TEST – JEE MAIN
Section – A
1. A circle touches the x-axis and also touches the circle with centre at (0, 3) and radius 2. The locus of the centre of the circle is
(1) an ellipse (2) a circle (3) a hyperbola (4) a parabola
2. The circle passing through the point (-1, 0) and touching the y-axis at (0, 2) also passes through the point____________
(1) 3 ,0
(2) 5 ,2
(3) 35 , 22
(4) (–4, 0)
3. Consider a circle C which touches the y-axis at (0, 5) and cuts off an intercept 32 on the x- axis. Then the centre and radius of the circle C are
49. A line y=mx+ 1 intersects the circle (x−3)2 + ( y + 2) 2 = 25 at the points P and Q. If the midpoint of the line segment PQ has x -coordinate 3 5 , then which one of the following options is correct? (2019 P1) (1) −3 ≤ m < −1 (2) 2 ≤ m < 4 (3) 4 ≤ m < 6 (4) 6 ≤ m < 8
50. Let the point B be the reflection of the point A (2, 3) with respect to the line 8x − 6y − 23 = 0. Let TA and TB be circles of radii 2 and 1 with centres A and B respectively. Let T be a common tangent to the circles TA and TB such that both the circles are on the same side of T. If C is the point of intersection of T and the line passing through A and B , then the length of the line segment AC is _____. (2019 P1)
(1) 118 ,5,5 2
(2) 118 ,5,10 2 ±
(3) 11859 5,, 22 ±
(4) 11859 ,5, 22 ±
4. The minimum radius of the circle which contains the three circles x 2 + y 2 – 4 y –5 = 0, x 2 + y 2 + 12 x + 4 y + 31 = 0 and x2 + y2 + 6x + 12y + 36 = 0
(1) 7 9003 28 +
(2) 845 4 9 + (3) 5 9493 36 +
(4) 5 9453 36 +
5. Let O be the centre of the circle x2 + y2= r2 , where 5 2 r > Suppose, PQ is a chord of this circle and the equation of the line passing through P and Q is 2x + 4y = 5. If the centre of the circumcircle of ∆OPQ lies on the line x + 2y = 9, then the value of r is
(1) 3 (2) 2
(3) 5 (4) 9
6. If one of the diameters of the circle 22 2262140xyxy+−−+= is a chord of the circle ()() 22 2 2222 xyr −+−= , then
(1) ()3/4 10 rr =
(2) ()7/8 10 rrr =
(3) ()3/8 10 rr =
(4) ()7/4 10 rrr =
7. The radius of the circle whose diameter is the common chord of the circles
x 2 + y 2 + 2 x + 2 y + 1 = 0 and
x2 + y2 + 4x + 3y + 2 = 0 is ___
(1) 2 5 (2) 1 5
(3) 1 (4) 17 2
8. Length of common tangent(s) of the circles
x2 + y2 = 6x, x2 + y2 + 2x = 0 is/are
(1) 2 2 fc
(2) 3,33
(3) 23
(4) 23,33
9. If the lines 2 x – 3 y + 6 = 0 and mx + 2y + 12 = 0 meets the coordinate axes at concyclic points then m = (1) 3 (2) –3 (3) 2 (4) –2
10. A variable circle which always touches the line x + y – 2 = 0 at point (1, 1) cuts circle
x2 + y2 + 4x + 5y − 4 = 0 at P and Q, then line PQ always passes through fixed point (1) (1, 1)
(2) (4, –2)
(3) (8, – 6)
(4) (0, 0)
11. Point on the circle x2 + y2 − 2x + 4y − 4 = 0 which is nearest to the line y = 2x + 11 is
−−+
(1) 63 1,2 55
(2) 63 1,2 55
(3) 63 1,2 55
(4) 63 1,3 55 +−−
12. The locus of the point of intersection of the tangents to the circle x 2 + y 2 = a 2 , which include an angle of 45° is the curve x2 + y2 = λa2. The value of λ equals to (where [.] denotes GIF)
(1) 2 (2) 4 (3) 6 (4) 16
13. The minimum length of the chord of the circle x 2 + y 2 + 2 x + 2 y − 7 = 0 which is passing through (1,0) is:
(1) 2 (2) 4
(3) 22 (4) 5
14. If the tangents AP and AQ are drawn from the point A(3,−1) to the circle x2 + y2 − 3x + 2y − 7 = 0 and C is the centre of circle, then the area of quadrilateral APCQ is:
(1) 9
(2) 4
(3) 2
(4) Non-existent
15. The distance between the chords of contact of tangents to the circle x2 + y2 + 2gx + 2fy + c = 0 from the origin and the point (g, f) is:
(1) 22gf +
(2) 22 2 gfc +−
(3) 22 22 2 gfc gf ++ +
(4) 22 22 2 gfc gf +− +
16. Radical centre of the circle drawn on the sides as a diameter of triangle formed by the lines 3x − 4y + 6 = 0 , x − y + 2 = 0 and 4x + 3y −17 = 0 is
(1) (3, 2)
(2) (3, –2)
(3) (2, –3)
(4) (2, 3)
17. The line y = x is tangent at (0,0) to a circle of radius 1. The centre of the circle is:
(1) either 11 -, 22
or 11 ,22
(2) either 11 , 22 or 11 , 22
(3) either 11 , 22 or 11 , 22
(4) either (1, 0) or (–1, 0)
18. The equation of the circle circumscribing the triangle formed by the points(3, 4)(1, 4)and (3, 2) is
(1) 8x2 + 8y2 −16x −13y = 0
(2) x2 + y2 − 4x − 8y+ 19 = 0
(3) x2 + y2 − 4x − 6y+ 11 = 0
(4) x2 + y2 − 6x − 6y+ 17 = 0
19. The equation of the tangent to the circle x 2 + y 2 − 4 x = 0 which is perpendicular to the normal drawn through the origin can be:
(1) x = 1
(2) x = 2
(3) x + y = 2
(4) x = 4
20. The line 2x − y + 1 = 0 is tangent to the circle at the point (2,5) and the centre of the circles lies on x − 2y = 4 . The radius of the circle is:
(1) 35
(2) 53
(3) 25
(4) 52
Section – B
21. The circle x 2 + y 2 − 4 x − 4 y + 4 = 0 is inscribed in a triangle which has two of its sides along the co-ordinate axes. If the locus of the circumcentre of the triangle is 22 0 xyxykxy +−++= . The value of k is _____.
22. Two circles touch the x–axis and the line y = mx. They meet at (9, 6) and at one more point and the product of their radii is 117 2 then value of m2 is _____.
23. Let A(-2, 2) and B(2, –2) be two points and AB subtends and angle of 45° at any points P in the plane in such a way that area of triangle APB is 8 square unit, then number possible position (s) of P is _____.
24. The number of lattice points that are interior to the circle x2 + y2 = 25 is ______.
25. If n (n ≥ 3) circles, the centre of no three circles are collinear. If the number of radical axes of the circles is equal to the number of radical centres of the circles then n2–4n–5 = ____.
CHAPTER TEST – JEE ADVANCE
2018 P2 Model
Section – A
[Numerical Value Questions]
1. Let S 1 = 0 and S 2 = 0 be two circles intersecting at P(6, 4) and both are tangent to x-axis and line y=mx (where m > 0). If product of radii of the circles S1 = 0 and S2 = 0 is 52 3 , then the value of m2 = ____ .
2. If circle x2 + y2 +2gx + 2fy + c =0 cuts each of the circles x2 + y2 −4 = 0, x2 + y2 − 6x − 8y + 10 = 0 and x 2 + y 2 + 2 x − 4 y − 2 = 0 at the extremities of diameter, then the value of (g2 + f2 − c) = ____
3. For how many values of p the circle x2 + y2 + 2x + 4y − p = 0 and the co-ordinate axes have exactly three common points ____
4. The centres of two circles C1 and C2 each of unit radius are at a distance of 6 units from each other. Let P be the mid point of the line segment joining the centres of C1 and C2 and C be a circle touching C1 and C2 externally. If a common tangent to C1 and C passing through P is also a common tangent to C2 and C1 , then the radius of the circle C is _____.
5. Two circles of radius 8 are placed inside a semi circle of radius 25. The two circles are each tangent to the diameter and to the semi circle. If the distance between the centres of the two circles is ‘ λ', then 5 5 λ+ is ______.
6. If p and q be the longest and the shortest distances respectively of the point (–7,2) from any point( a,b ) on the circle x 2 + y 2 − 10x − 14y − 51 = 0. If s is the geometric mean of p and q, then 11 s is equal to _____.
7. Two parallel chords of a circle of radius 2 are at a distance 31 + apart. If the chords subtend at the centre, angles of π/k and 2π/k where k > 0, then the value of [ k ] is ____. where [.]=GIF
8. If the circles x2 + y2 + (3+sinθ)x + 2cos ϕ y = 0 and x2+y2 + (2cosϕ)x + λy = 0 touch each other, then the maximum value of λ is
Section – B
[Multiple Option Correct MCQs]
9. Let RS be the diameter of the circle x2 + y2 = 1, where S is the point (1, 0).Let P be a variable point (other than R and S) on the circle and tangents to the circle at S and P meet at the point Q. The normal to the circle at P intersects a line drawn through Q parallel to RS at point E. Then the locus of E passes through the point(s)
(1) 11 , 3 3
(3) 11 , 3 3
(2) 11 , 42
(4) 11 , 42
10. Equation(s) of circle(s) concentric with x2 + y2 + 2x + 4y = 0 and touching the circle
x2 + y2 − 2x − 1 = 0 is/are
(1) x2 + y2 + 2x + 4y + 3 = 0
(2) x2 + y2 + 2x + 4y – 13 = 0
(3) x2 + y2 + 2x + 4y – 3 = 0
(4) x2 + y2 + 2x + 4y + 13 = 0
11. Let T be the line passing through the points P (-2, 7) and Q (2, –5). Let F 1 be the set of all pairs of circles( S 1, S 2) such that T is tangent to S 1 at P and tangent to S 2 at Q, and also such that S 1 and S 2 touch each other at a point, say, M. Let E1 be the set representing the locus of M as the pair (S1, S2) varies in F1 . Let the set of all straight line segments Joining a pair of
distinct points of E1 and passing through the point R (1, 1) be F2 . Let E2 be the set of the mid-points of the line segments in the set F 2. Then, which of the following statement(s) is (are) TRUE?
(1) The point (−2, 7) lies in E1
(2) The point 47 , 55 does not lies in E2
(3) The point 1 ,1 2 lies in E2
(4) The point 3 0, 2 does not lies in E1
12. Circles x2 + y2 = 1 and x2 + y2 – 8x + 11 = 0 cut off equal intercepts on a line through the point (–2, 1/2) the slope of the line, is
(1) 129 14 −+ (2) 17 4 + (3) 129 14 (4) 17 4
13. If θ is the angle subtended at P(x1,y1) by the circle S ≡ x2 + y2 + 2gx + 2fy + c = 0, then (Where 22 11111 22 Sxygxfyc =++++ ).
(1) 1 22 cot S gfc θ = +−
(2) 1 22 cot 2 S
14. The equations of four circles are (x ± a)2 + (y ± a)2 = a2. The radius of a circle touching all the four circles is
(1) () 21 a (2) 22a (3) () 21 a + (4) () 22 a +
Section
– C
[Single Option Correct MCQs]
15. If a > 2b > 0 then the positive value of m for which 2 1 ymxbm =−+ is a common tangent to x2 + y2 = b2 and (x − a)2 + y2 = b2 is
(1) 22 2 4 b ab
(2) 22 4 2 ab b (3) 2 2 b ab
(4) 2 b ab
16. A tangent PT is drawn to the circle x2 + y2 = 4 at the point () 3,1 P . A straight line L, perpendicular to PT is a tangent to the circle (x − 3)2+y2 = 1. A possible equation of L is (1) 31xy−= (2) 31xy+= (3) 31xy−=− (4) 35xy+=
17. A circle of radius 1 is inscribed in an equilateral triangle PQR. The points of contact of C with the sides PQ, QR, RP are D, E, F, respectively. The line PQ is given by the equation 360 xy+−= and the point D is 333 , 22 . Further, it is given that the origin and the centre of C are on the same side of the line PQ. The equation of circle C is
(1) ()() 2 2 2311xy−+−=
(2) () 2 2 1 23 1 2 xy −++=
(3) ()() 2 2 311xy−++=
(4) ()() 2 2 311xy−+−=
18. Let S be the circle in xy play defined by the equation x 2 + y 2 = 4. Let P be a point on the circle S with both coordinates being positive. Let the tangent to S at P intersect the coordinate axes at the point M and N.
ANSWER KEY
– II
Then, the mid-point of the line segment MN must lie on the curve
Studying the parabola, a fundamental conic section, is vital in mathematics and physics. Its application ranges from describing projectile motion to designing satellite dishes. Understanding its properties enables solving real world problems, making it essential in fields like engineering, astronomy, and optics.
14.1 CONIC SECTION
In tw o-dimensional geometry, figures like circle, parabola, ellipse, hyperbola, pair of straight lines, straight line, and point are called conic sections, because each is a section of a double napped right circular cone with a plane as shown in the below figures.
Suppose, the cutting plane makes an angle β with the axis of the cone and suppose the generating angle of the cone is α . Then, the section is
i. a circle, if 2 β=π
PARABOLA CHAPTER 14
ii. an ellipse, if 2 α<β<π
iii. a parabola, if α=β
iv. a hyperbola, if 0 ≤β<α
We get the degenerated sections as below, when the plane passes through the vertex of the cone.
i. A point when 2 α<β<π
ii. A straight line when α=β
iii. A pair of straight lines when 0 <β<α
The locus of a point which moves in a plane so that its distance from fixed point and from fixed line are in constant ratio is called conic
The constant ratio is called eccentricity e , fixed point is called focus, and fixed line is called directrix of conic.
i. If e = 1, then the conic is a parabola.
ii. If 0 < e < 1, then the conic is an ellipse.
iii. If e > 1, then the conic is a hyperbola.
iv. If e = 0, then the conic is a circle.
v. If e is more than a bigger value, then the conic is a pair of straight lines.
14.1.1 General Equation of Conic Section
If the focus is S () , αβ and the equation of directrix is ax + by + c = 0, then the equation of the conic section whose eccentricity e is SP = e PM
The general equation of conic is 222220 axhxybygxfyc +++++=
Identification of the conic: Consider the equation,
222220Saxhxybygxfyc ≡+++++= , such that 222 20abcfghafbgch+−−−≠
i. If 2 hab = , then S = 0 represents a parabola
ii. If 2 hab < , then S = 0 represents an ellipse.
iii. If 2 hab > , then S = 0 represents a hyperbola.
iv. If 2 hab > , where a + b = 0, then S = 0 represents a rectangular hyperbola.
Vocabulary:
i. A line passing through the focus of a conic and perpendicular to the directrix is called principal axis of the conic. A parabola will have only one axis, while an ellipse and hyperbola each have two axes.
ii. The axis of parabola is called principal axis, axes of ellipse are called major axis and minor axis, and axes of hyperbola are called transverse axis and conjugate axis.
iii. The points of intersection of conic and its axes are called vertices of conic. A conic has at most two vertices.
iv. Centre of the conic is midpoint of the line segment joining the vertices.
v. Centre of the conic
222220Saxhxybygxfyc ≡+++++= is 22 , hfbgghaf abhabh
, where ab ≠ h2
vi. Any chord passing through the focus is called focal chord.
vii. A focal chord, which is perpendicular to the axis, is called the latus rectum.
viii. A chord passing through any point on the conic and perpendicular to the axis is called double ordinate.
ix. The distance of any point on the conic from focus is called focal distance of that point.
1. Identify the conic represented by the equation 22 13183721420 xxyyxy−+++−=
Sol. Comparing the given equation with 222220 axhxybygxfyc +++++= , we have 13,9,37,1,7,2ahbgfc ==−====−
96212663737162 16000 abcfghafbgch ∆=+−−−
2 981,481hab=−==
Clearly, 0 ∆≠ and 2 hab < Hence, the given equation represents an ellipse.
Try yourself:
1. Find the equation of the conic whose focus is (–2,3) and the directrix is 2x + 3y – 4 = 0, having eccentricity 1.
Ans: 22 912468541530 xxyyxy−++−+=
TEST YOURSELF
1. The equation 16x2+y2+8xy–74x –78y+212 = 0 represents (1) a circle (2) a parabola (3) an ellipse (4) hyperbola
2. If ax 2 +4 xy + y 2 + ax +3 y +2 = 0 represents a parabola, then a = (1) –4 (2) 4 (3) 0 (4) None
3. Find the equation of the parabola whose focus is at (–1, –2) and the directrix is the straight line x–2y+3 = 0.
(1) 4x2+y2+4xy+4x+32y+16 = 0
(2) 4x2+y2+4xy = 0
(3) x2+y2+4xy+x+8y = 0
(4) None of these
4. An equation of the parabola, whose focus is (–3, 0) and the directrix x+5 = 0, is
(1) y2 = 4(x+5) (2) y2 = 4(x+4)
(3) y2 = 4(x+3) (4) y2 = 4(x–3)
Answer Key
(1) 2 (2) 2 (3) 1 (4) 2
14.2 EQUATION OF PARABOLA
The locus of a point which moves in a plane so that its distances from a fixed point (focus) is equal to its distance from a fixed straight line (directrix) is called parabola.
14.2.1 Equation of Parabola Whose Axis Is along Coordinate Axes
Equation of parabola in the standard form is 2 4 yax = . This is called simplest form of the equation of a parabola. This parabola open right side having directrix perpendicular to x axis, axis is along x axis, and vertex at origin.
For the parabola 2 4;0yaxa=> ,
i. vertex is () 0,0 A
ii. focus is () ,0Sa .
iii. equation of the directrix is 0 xa+= .
iv. equation of the principal axis is 0 y =
v. equation of tangent at the vertex is 0 x =
vi. length of the latus rectum is 4a
vii. equation of latus rectum is 0 xa−=
viii. ends of the Latus rectum are ()() ,2,,2 aaaa
ix. focal distance of a point () 11 , Pxy is 1 xa +
Nature of the curve 2 4 yax = , a0 > : L Latus rectum
Focal chord S X A Y’ X’ L’ Z Y
directrix
Double ordinate
i. The curve y 2 = 4 ax is symmetric about the x-axis. It is principal axis of parabola.
ii. The curve meets the x-axis at only point, vertex (0, 0).
iii. The curve lies in the first and fourth quadrants.
iv. y-axis touches the parabola at origin.
Equation of the parabola in the form of 2 4 yax =− , 0 a > : This parabola open left side having directrix perpendicular to x axis, axis is along x axis, and vertex at origin L l X’ Y Y’
i. vertex is () 0,0 A
ii. focus is () ,0Sa
iii. equation of the directrix is 0 xa−=
iv. equation of the principal axis is 0 y =
v. equation of tangent at the vertex is 0 x =
vi. length of the latus rectum is 4a
vii. equation of latus rectum is 0 xa+=
viii. ends of the latus rectum are ()() ,2,,2 aaaa
ix. focal distance of a point () 11 , Pxy is 1 xa
Equation of the parabola in the form of
2 4;0xaya=> : This parabola open upward having directrix perpendicular to y-axis, axis is along y-axis, and vertex at origin. L X Y y=0 L’
S(0,a)
A(0,0) y = a=0 x=0
Z(0,–a)
For the parabola 2 4;0xaya=> ,
S(–a,0) Z X
(–a, –2a) (–a, 2a)
A (0,0) (a,0) X–a=0 X=0
For the parabola 2 4;0yaxa=−> ,
i. vertex is () 0,0 A
ii. focus is () 0, Sa
iii. equation of the directrix is 0 ya+=
iv. equation of the principal axis is 0 x =
v. equation of tangent at the vertex is 0 y =
vi. length of the latus rectum is 4a
vii. equation of latus rectum is 0 ya−=
viii. ends of the latus rectum are ()() 2,,2, aaaa
ix. focal distance of a point () 11 , Pxy is 1 ya +
Equation of the parabola in the form of 2 4 xay =− , a > 0: This parabola open downward having directrix perpendicular to y-axis, axis is along y-axis, and vertex at origin. L Y X y=0 y–a=0 Z(0,a)
(–2a, –a) (2a, –a) A(0,0) x=0 S(0,–a)
(0,–a)
For the parabola 2 4;0xaya=−> ,
i. vertex is () 0,0 A
ii. focus is () 0, Sa
iii. equation of the directrix is 0 ya−=
iv. equation of the principal axis is 0 x =
v. equation of tangent at the vertex is 0 y =
vi. length of the latus rectum is 4a
vii. equation of latus rectum is 0 ya+= viii. ends of the latus rectum are ()() 2,,2, aaaa
ix. focal distance of a point () 11 , Pxy is 1 ya
2. If the equation of the parabola is 2 yax = and passing through the point (1, 2), then find the equation of the directrix.
Sol. Since the parabola 2 yax = is passing through the point(1, 2), 4 = a , the equation of the parabola is 2 4.yx = this is standard parabola, having axis along x-axis and vertex is at origin. And a = 1
Therefore, the equation of the directrix is x + 1 = 0.
Try yourself:
2. If the length of a double ordinate of 2 8 yx = is 24 units, then what is the equation of the double ordinate?
Ans: x180 −=
14.2.2 Parabolas Having Axis Parallel to Coordinate Axes
Equation of parabola in the form of ()() 2 4;0ykaxha −=−> : This parabola open right side having directrix perpendicular to x-axis, axis is parallel to x-axis, and vertex is not at origin.
For the parabola ()() 2 4;0,ykaxha −=−>
i. vertex is () , Ahk
ii. focus is () , Shak +
iii. equation of the directrix is xha −=−
iv. equation of the principal axis is yk =
v. equation of tangent at the vertex is xh =
vi. length of the latus rectum is 4 a
vii. equation of latus rectum is xha −=
viii. ends of the latus rectum are ()() ,2,,2 ahkaahka +−++
Equation of parabola in the form of ()() 2 4;0xhayka −=−> : This parabola open upward having directrix perpendicular to y-axis, axis is parallel to y-axis, and vertex is not at origin.
vi. length of the latus rectum is 4a
vii. equation of latus rectum is yka −=
viii. ends of the latus rectum are ()() 2,,2, ahkaahka ++−++
Equation of parabola whose axis is i. parallel to x-axis is of the form
2 xaybyc =++
ii. parallel to y-axis is of the form
2 yaxbxc =++
The length of latus rectum of the parabola
2 0 axbxmyn+++= is m a
The length of latus rectum of the parabola
2 0 aybymxn+++= is m a .
Characteristics of the Parabola 2 yaxbxc =++
For the parabola ()() 2 4;0,xhayka −=−>
i. vertex is () , Ahk
ii. focus is () , Shka +
iii. equation of the directrix is yka −=−
vi. equation of the principal axis is xh =
v. equation of tangent at the vertex is yk =
i. Vertex is 2 4 , 24 bacb aa
ii. Focus S is 2 41 , 244 bacb aaa
iii. Equation of the directrix is 2 41 0 44 bac y aa ++=
iv. Equation of the principal axis is 0 2 b x a +=
v. Equation of tangent at the vertex is 2 4 0 4 bac y a +=
vi. Length of the latus rectum is 1 a
vii. Equation of the latus rectum is 2 41 0 44 bac y aa +−=
3. Find the equation of parabola whose vertex is (1, –2) and slope of axis is undefined and Latus rectum 4.
Sol. Given: The slope of axis is undefined, means axis is y-axis.
Since the length of the Latus rectum is 4 , 44. a =
If the parabola open upward, then the equation of the parabola is ()() 2 142xy−=+
If parabola open downward, then the equation of the parabola is ()() 2 142xy−=−+
Try yourself:
3. Find the equation of the parabola whose vertex is () 2,5 and focus is ()2,2. Ans: )()( 2 2125xy−=−−
14.2.3 Oblique Parabola
The equation of parabola having focus at () , αβ and directrix 0 lxmyn++= is ()()()2 22 22 lxmyn xylm ++ −α+−β= +
This is an oblique parabola; its axis is not parallel to any of coordinate axes. Its axis is ()() 0 mxly−α−−β= .
For any parabola, suppose that S is focus, L is directrix, and Z is foot of the perpendicular of S on L . Then, the
i. vertex A is midpoint of SZ
ii. length of the latus rectum is () 2 SZ
iii. axis is a line perpendicular to directrix L and passing through vertex A , focus S and Z
iv. tangent at vertex is a line parallel to directrix and passing through vertex.
If 0, lxmyn++= 0 mxlyk−+= are tangent at vertex, axis of the parabola then the equation of the parabola is 222
()(4)|| mxlykalmlxmyn −+=+++
Notation: Here, after, the following notations will be adopted throughout this chapter:
i. 2 4 Syax =−
ii. ()111 2 Syyaxx =−+
iii. 2 1111 4 Syax =−
iv. ()121212 2 Syyaxx =−+
4. If the focus is () 1,1 and directrix is the line 290xy+−= , then find the vertex of the parabola.
Sol. Find the foot of the perpendicular of focus on the directrix.
Let () , hk is foot of the perpendicular of () 1,1 on the line 290xy+−= () 129 11 2 1214 hk−+ === +
Simplifying, 3,3hk==
Hence, the foot of the perpendicular of () 1,1 on 290xy+−= is () 3,3
Since, the vertex is midpoint of () 1,1 and ()3,3. It is equal to () 2,1
Therefore, the vertex of the parabola is () 2,1
Try yourself:
4. If the equation 22 16874782120 xkyxyxy++−−+= represents a parabola, then find the value of k.
Ans:1
14.2.4 Position of a Point with Respect to Parabola
A parabola divides the plane into three regions. The region containing the focus is called the interior of the parabola. The region consisting of the curve itself. The region containing the remaining portion of the plane is called the exterior of the parabola.
Let () 11 , Pxy be any point in the plane of parabola S = 0. Then,
i. () 11 , Pxy lies outside the parabola ⇔
11 0 S > .
ii. () 11 , Pxy lies on the parabola ⇔
11 0 S = .
iii. () 11 , Pxy lies inside the parabola ⇔
11 0 S < .
5. The point () 2,aa lies inside the region bounded by the parabola 2 4 xy = and its latus rectum. Then, find the range of a
Sol. The point () 2,aa lies inside 2 4 xy =
Hence, () 2 44010 aaaa−<⇒−<
So, () 0,1 a ∈
Equation of the Latus rectum is 1 y = , so that the point () 2a,a lies below the line 1 y =
Hence, 1 a <
Therefore, () 2,aa lies inside the region bounded by the parabola 2 4 xy = and its latus rectum, when () 0,1 a ∈
Try yourself:
5. The point () 2,1 mm −+ is an interior point of the smaller region bounded by the circle 22 4 xy+= and the parabola 2 4 yx = . Then, find the range of m
Ans: 1526 m −<<−+
14.2.5 Parametric Equations of Parabola
The parametric equations of parabola 2 4 yax = are 2 ,2 xatyat == . Any point on the parabola ,
2 4 yax = can be taken asP () 2 ,2 atat . This can be represented as P( t).
The parametric equations of parabola 2 4 xay = are 2 2, xatyat ==
6. Find the vertex of the parabola whose parametric equations 2 1 xtt=−+ and 2 1 ytt=++ where tR ∈
Sol. () 2 21,2 xytyxt +=+−=
Eliminating t, we get, the equation of parabola is
()() 2 22yxxy −=+−
Try yourself:
6. Find the equation of the parabola described parametrically by 2 52,104.xtyt=+=+
Ans: )()( 2 4202yx−=−
TEST YOURSELF
1. If the focus is (1,–1) and directrix is the line x+2y–9 = 0, then vertex of the parabola is (1) (1, 2) (2) (2, 1) (3) (1, –2) (4) (2, –1)
2. Given below are two statements:
Statement I : For the parabola
y 2 +6 y –2 x =–5, the vertex is (–2,–3)
Statement II : For the parabola
y 2 +6 y –2 x =–5, the directrix is y+3 = 0.
In light of the above statements, choose the correct answer from the options given below
(1) Both statement I and statement II are correct.
(2) Both statement I and statement II are incorrect.
(3) Statement I is correct but statement II is incorrect.
(4) Statement I is incorrect but statement II is correct.
3. Equation of parabola whose vertex is (2, 5) and focus (2, 2) is
(1) (x–2)2 = 12(y–5) (2) (x–2)2 = –12(y–5)
(3) (x–2)2 = 12(y–2) (4) (x–2)2 = –12(y–2)
4. Equation of parabola whose vertex is (–1, 2) and focus is (3, 2) is
(1) (y–2)2 = 17(x–3)
(2) (y–2)2 = 16(x–3)
(3) (y–2)2 = 16(x+1)
(4) (y–2)2 = –16(x+1)
5. Equation of parabola having vertex (–1,–2), whose axis is vertical and which passes through (3, 6), is
(1) x2+2x–2y–3 = 0
(2) x2+4x–8y–13 = 0
(3) x2+4x–16y–12 = 0
(4) x2+4y+16y+12 = 0
6. The parabola y = ax2+2ax+b is symmetric about the line
(1) 2ax+b = 0 (2) ax+1 = 0
(3) x+1 = 0 (4) x = 0
7. If the vertex of parabola y = x2– 8x + c lies on x-axis, then it is
(1) (2, 0) (2) (4, 0) (3) (6, 0) (4) (8, 0)
8. The vertex of the parabola x2+8x+12y+4 = 0, is
(1) (–4, 1) (2) (4, –1) (3) (–4, –1) (4) (4, 1)
9. The equation of the Latus rectum of the parabola x2–12x–8y+52=0, is (1) x = 4 (2) y = 4 (3) x = 6 (4) y = 2
Chords on a parabola are line segments connecting any two points on the curve. When extended as lines, as one point approaches the other along the curve, the chord becomes a tangent to that point on the curve. Normals are perpendicular lines to tangents at their point of contact.
14.3.1 Chord
1. Equation of chord
i. The equation of the chord joining two points () 11 , Pxy and () 22 , Qxy in the parabola 0 S = is 1212SSS +=
ii. The equation of the chord of the parabola 2 4 yax = joining 12 , tt is () 12 12 22 yttxatt +=+
2. Focal chord:
i. The condition for the chord joining 12 , tt to be focal chord is 12 1 tt =−
ii. The length of the focal chord passing through the point t is 2 1 at t
+
iii. If one end of focal chord is () 2 ,2 atat , then the other end is 2 2 , aa tt
iv. If () 11 , xy and () 22 , xy are ends of focal chord, then 2 12 , xxa = and 2 12 4 yya =−
v. If a focal chord makes an angle θ with x-axis, then the length of the focal chord is 2 4csc a θ
vi. If PSQ is a focal chord and l is the length of semi Latus rectum of parabola 2 4 yax = , then ,, SPlSQ are in HP. It means 112 SPSQl += or 111 SPSQa +=
3. The end points of a double ordinate are () 2 ,2 atat , () 2 ,2 atat . The length of double ordinate is 4at . Its equation is 2 xat =
4. In parabola 2 4 yax = , if a chord makes an angle θ with x-axis, and passing through the vertex, then the length of the chord is 2 4coscsc a θθ .
5. Length of double ordinate of the parabola 2 4 yax = which subtends an angle θ at the vertex is 8cot 2 a θ .
6. An equilateral triangle inscribed in the parabola 2 4 yax = , whose vertex is at the vertex of the parabola then the length of the side is 83a .
7. In parabola 2 4 xay = , if a chord makes an angle θ with x- axis, and passing through the vertex, then the length of the chord is 2 4sinsec a θθ
8. If an equilateral triangle is inscribed in the parabola 2 4 xay = , whose vertex is at the vertex of the parabola, then the length of the side is 83 a .
9. The length of double ordinate of the parabola y2 = 4ax at a distance of p units from the vertex is 4 ap
7. If an equilateral triangle is inscribed in a parabola 2 12 yx = , with one of the vertices at the vertex of the parabola, then what is its height? Sol. Y M B x A O 2 3 cot30 62 23 OMtt AMt t °=== ⇒=
Height =OM= 2 336 t =
Try yourself:
7. What is the length of the chord intercepted by the parabola 2 3 yxx =+ on the line 5? xy+=
Ans: 62
14.3.2 Tangent
The equation of tangent to the parabola 0 S = at () 11 , Pxy is 1 0. S =
1. The equation of tangent to the parabola 2 4 yax = at () 11 , Pxy is ()11 2 yyaxx =+ i. The slope of the tangent to the parabola 2 4 yax = at () 11 , Pxy is
= and y intercept is
ax c y =
ii. The slope of tangent to the parabola
2 4 yax = at t is 1 t
2. The equation of the tangent to the parabola
2 4 yax = at t is 2 0 ytxat−−=
3. The condition for the line ymxc =+ to be a tangent to the parabola 2 4 yax = is a c m = and the point of contact is
2 2 , aa mm
.
4. The condition for the line 0 lxmyn++= to be a tangent to the parabola 2 4 yax = is 2 ln am = . Point of contact is 2 , nam ll
5. The condition for the line ymxc =+ to be a tangent to the parabola 2 4 xay = is 2 cam =− and the point of contact is () 2 2,amam
6. The condition for the line 0 lxmyn++= to be a tangent to the parabola 2 4 xay = is 2 n mal = . Point of contact is 2 , aln mm
7. The slope form of tangent to the parabola 2 4 yax = is a ymx m =+
8. If m is the slope of tangent to the parabola 2 4 yax = at () 11 , Pxy , then the equation of tangent is 2 11 0 mxmya−+=
9. The condition for the line ymxc =+ to be a tangent to the parabola () 2 4 yaxa =+ is 1 cam m =+
10. The slope form of the tangent to the parabola () 2 4 yaxa =+ is () a ymxa m =++
11. If the tangents drawn from P to the parabola () 2 4 yaxa =+ and () 2 4 ybxb =+ are right angles, then the locus of the point P is 0 xab++= .
12. Equation of common tangent to the parabolas 2 4 yax = , 2 4 xby = is () 11 2 33 3 0 axbyab++=
13. Locus of foot of perpendicular from focus on any tangent is tangent at the vertex of the parabola.
Q P(at2,2at) S(a, 0) x=–a x O y
14. Image of focus with respect to any tangent lies on directrix.
15. Circle drawn on focal length as diameter touches tangent at vertex. P(at2, 2at) S(a, 0) O x y
16. Circle drawn on focal chord as diameter touches the directrix.
P(at2, 2at) x =–a S(a,0) Q(a/t2, –2a/t) x
17. Area of the triangle inscribed in parabola y 2 = 4 ax is 122331 1 ()()(), 8 yyyyyy a where y1, y2, y3 are ordinates of the vertices
18. Area of the triangle formed by the tangents drawn at three points on the parabola y2 = 4ax, whose ordinates are y1, y2, y3 is 122331 1 ()()(). 16 yyyyyy a
19. The orthocentre of the triangle formed by the tangents drawn at the points 123 ,, ttt to the parabola 2 4 yax = is (() ) 123123 , aatttttt −+++
20. The orthocentre of the triangle formed by the tangents drawn at the points 123 ,, ttt to the parabola 2 4 yax = lies on the directrix.
21. Circumcircle of triangle formed by three tangents to the parabola always passes through focus of parabola.
8. If the equation of the tangent to the parabola y2=8x which is parallel to the line x–y+3=0 is x–y+k=0 then find the value of k.
Sol. Slope of the line x–y+3=0 is1.
Slope of required tangent m=1 For the parabola y2=8x,
equation of the tangent parallel to x–y+3=0 is 2 20 1 a ymxyxxy m =+⇒=+⇒−+=
On comparison, k=2
Try yourself:
8. Find the slope of line that touches both y2 = 4x, x2 = –32 y.
Ans : 1/2
14.3.3 Normal
The equation of normal to the parabola 2 4 yax = at () 11 , Pxy is ()() 111 20xxyyya −+−= ,
1. The condition for the line ymxc =+ to be normal to the parabola 2 4 yax = is c = –2am–am3 .
2. The slope form of the normal to the parabola 2 4 yax = is 3 2 ymxamam =−− .
3. The condition for the line 0 lxmyn++= to be normal to the parabola 2 4 yax = is 32220alalmmn++=
4. The equation of normal to the parabola 2 4 yax = at () 2 ,2 atat is 3 2 yxtatat +=+
i. The slope of normal to the parabola 2 4 yax = at t is t
ii. If the normal to the parabola at t1 intersect the parabola again at 2 t , then 21 1 2 tt t =−−
iii. If the normal to the parabola at t subtends a right angle at the focus, then 2 t =± .
vi. If the normal to the parabola at t subtends a right angle at the vertex, then 2 t =± .
v. The length of the normal chord to the parabola 2 4 yax = at t is () 3 2 2 2 41at t +
5. For the parabola 2 4 yax = , if the normal at 1t and the normal at 2 t intersect on the parabola at t 3 , then 12 2 tt = , t1 + t2 + t3 = 0.
6. The point of intersection of normals drawn at 12 , tt to the parabola 2 4 yax = is (()() ) 2 2,12121212 aattttatttt ++−−+
7. Number of normal lines at () 11 , Pxy :
i. Substituting () 11 , Pxy in 3 2,yxtatat +=+ we get () 3 11 20attaxy+−−= , it’s a cubic equation. It has at most three real roots and at least one real root, so that the number of normals to the parabola at any point is at least one and at most 3.
ii. Number of normals to the parabola at () 11 , Pxy does not depend on the position of () 11 , Pxy , it depends on the number of distinct roots of the equation () 3 11 20attaxy+−−=
8. If 123 ,, ttt are the roots of the equation () 3 11 20attaxy+−−= , then 123 ,, ttt are called feet of the normals drawn from a point () 11 , Pxy
i. 123 0 ttt++=
ii. 1 122331 2ax tttttt a ++=
iii. 1 123 ttty a =
iv. If 123 ,, ttt are feet of the normals drawn from P to the parabola, then the circle passing through the points 123 ,, ttt is also passing through vertex of the parabola.
v. Area of the triangle formed by feet of the normals 123 ,, ttt is () () () 2 122331 atttttt .
vi. Area of the triangle formed by three normals drawn to the parabola 2 4 yax = is () () ()()2 2 122331123 1 2 attttttttt =−−−++
vii. The centroid of the triangle formed by the feet of the 3 normals to the parabola 2 4 yax = from a given point lies on the axis of the parabola.
9. The normals drawn at the ends of the Latus rectum intersect at right angles and the point of intersection is () 3,0 a .
10. The normals and tangents drawn at the end points of the Latus rectum of 2 4 yax = forms a square of area 8 a2 .
11. The tangent at one end of the focal chord is parallel to the normal at the other end of the focal chord.
12. If three normals are drawn to parabola 2 4 yax = from a point (h,0) on the principal axis then 2 ha >
13. The circle passing through three feet of the normals drawn from a point to the parabola, passes through the vertex of the parabola.
14. The length of the normal chord of the parabola is least when it subtends a right angle, at the vertex.
9. Find the number of distinct normals that can be drawn from 111 , 44 to the parabola 2 4.yx =
Sol. Equation of normal to the parabola 2 4 yx = is 3 2 ymxmm =−−
Therefore, number of distinct normals that can be drawn=2
Try yourself:
9. If y = x + 2 is a normal to the parabola y2 = 4ax, then find the value of a Ans: 1 a 3 =
TEST YOURSELF
1. If the line 2 y = 5 x + k is a tangent to the parabola y2 = 6x, then k = (1) 2 3 (2)
(3) 3 5 (4)
2. The line x+y = k touches the parabola y = x–x2, if k = (1) 0 (2) –1 (3) 1 (4) 2
3. Point of contact of y = 1– x w.r.t y2 –y+x = 0, is (1) (1, 1)
(2) 11 , 22
(3) (0, 1) (4) (1, 0)
4. The condition that the line y = mx + c is a tangent to parabola y2 = 4a(x+a) is
(1) 1 cam m =+
(2) 1 cam m
=−
(3) a c m = (4) 1 am m
=−
5. If 5x –2 y + k =0 is a tangent to the parabola y2 = 6x, then their point of contact is.
(1) 66 , 55
(3) 66 , 255
(2) 66 , 525
(4) 66 , 2525
6. The equation of normal to the curve x2 = 4y at (2, 1) is
(1) 2x+y+4 = 0
(2) x+y–3 = 0
(3) 2x–y+4 = 0
(4) 2x–y–4 = 0
7. The equation to the normal to the parabola x2 = 2y at (1,2) is
(1) x+y-3 = 0
(2) x–y+6 = 0
(3) x–y+5 = 0
(4) x–y+4 = 0
8. If the line 2x+y = k is a normal to y2 = 12x, then k =
(1) 12 (2) –12
(3) 36 (4) –36
9. The condition for the line lx+my+n = 0 to be a normal to y2 = 4ax is
(1) al2+2alm2+m2n = 0
(2) al3+2alm2+m2n = 0
(3) al2+2alm2+m2n2 = 0
(4) al3+alm2+m2n = 0
Answer Key
(1) 4 (2) 3 (3) 3 (4) 1
(5) 3 (6) 2 (7) 1 (8) 3
(9) 2
14.4 PAIR OF TANGENTS
The combined equation of pair of tangents drawn from a point () 11 , Pxy to the parabola
S0 = is 2 SSS111 =⋅
1. The slope form of a tangent to the parabola 2 4 yax = is 2 0 mxmya−+=
2. If 12 , mm are the slopes of tangents to the parabola drawn from an external point ()11,,Pxy then they are the roots of the equation 2 11 0 mxmya−+= . Hence, 1 12 1 y mm x += and 12 1 a mm x = .
3. If θ is acute angle between the pair of tangents drawn from () 11 , Pxy to the parabola S0 = , then 11 1 tan S xa θ= +
4. The locus of the point of intersection of perpendicular tangents is 0 xa+= , directrix of the parabola.
5. The angle between the tangents drawn from any point on the directrix to the parabola is 90° .
6. The angle between the tangents drawn at the ends of the latus rectum is 90° and point of intersection of these tangents is () ,0a
7. The point of intersection of tangents to the parabola 2 4at,12 yaxtt = is (() ) 1212 , attatt +
8. Let the tangents drawn to 2 4 yax = ()() 22 at,2,,21122 PatatQatat at T then i. The line joining T and midpoint of PQ is parallel to axis.
ii. If SPQ is focal chord, then STPQ ⊥ or focus S is foot of perpendicular of T on PQ.
10. Two tangents are drawn from the point (-2, –1) to the parbola 2 4 yx = . If θ is the angle between the tangents, then find the value of tan θ .
Try yourself:
10. If 12 , mm are the slopes of the tangents drawn to the parabola 2 4,yx = then find the value of 12 11 . mm + Ans: 3 2
TEST YOURSELF
1. Two tangents are drawn from (–2,–1) to y2 = 4 x. If θ is the acute angle between two tangents, then tan θ is equal to (1) 3
2. Angle between tangents drawn from origin to parabola y2 = 4a(x–a) is (1) 2 π (2) 3 π (3) 4 π (4) 6 π
3. The angle between the tangents drawn from the point (1, 4) to the parabola y2 = 4x is (1) 6 π (2) 4 π (3) 3 π (4) 2 π
4. The angle between the tangents drawn from the point (3,4) to the parabola y2–2y+4x = 0, is
(1) () 1 tan85/7
(2) () 1 tan12/5
(3) 5 ,0 2
(4) none of these
5. Equations of two tangents that can be drawn from (3,4) to the parabola y2 = 4x are,
(1) y = x+1, 3y = x+9
(2) y = x–1, 3y = x–9
(3) y = –x–1, 3y = –x–9
(4) –y = x+1, –3y = x+9
6. PA and PB are the tangents drawn to y2 = 4ax from point P. These tangents meet the y-axis at the points A1 and B1, respectively. If the area of triangle PA1B1 is 2 sq. units, then locus of P is
(1) (y2 + 4ax)x2 = 8
(2) (y2 + 4ax)x2 = 16
(3) (y2 – 4ax)x2 = 8
(4) (y2 – 4ax)x2 = 16
Answer Key
(1) 1 (2) 1 (3) 3 (4) 1 (5) 1 (6) 4
14.5 CHORD OF CONTACT, MIDPOINT OF CHORD
1. The equation of chord of contact of the point () 11 , Pxy with respect to the parabola
S0 = is 1 S0 = .
2. The length of the chord of contact of () 11 , Pxy with respect to the parabola
2 4 yax = is () 22 111 1 4 Sya a +
3. Area of the triangle formed by the tangents from () 11 , Pxy and chord of contact is () 3 2 11 2 S a .
4. The equation of the chord having () 11 , Pxy as its midpoint to the parabola 0 S = is 111SS = .
5. The length of the chord of 2 4 yax = having () 11 , xy as its midpoint is () () 22 111 1 4 Sya a −+
6. The tangent at any point on the parabola bisects the angle between the focal distance of the point and the perpendicular on the directrix from the point.
7. If tangents at P and Q meet at T , then i. TP and TQ will subtend equal angle at focus S ii. ()2 STSPSQ =⋅
iii. the triangles STP and STQ are similar
8. If N is foot of perpendicular from focus S on tangent at point P to the parabola, then N lies on tangent at vertex and ()2 SNSASP =⋅ (where A is vertex).
9. If tangents at point Q and R intersect at P. If 123 ,, ppp are lengths of perpendiculars from ,, PQR , respectively, on any tangents to parabola, then 123 ,, ppp are in GP.
10. The area of the triangle inscribed in a parabola 2 4 yax = is () () ()122331 1 , 8 yyyyyy a where
123 ,, yyy are ordinates of angular points on the parabola.
11. The area of the triangle formed by the tangents to the parabola 2 4 yax = at three points whose ordinates are 123 ,, yyy is () () ()122331 1 16 yyyyyy a
12. The portion of tangent to a parabola intercepted between point of contact and the directrix subtends a right angle at focus.
13. If the tangent at any point P of a parabola intersects the axis of parabola at any point T and M is the foot of perpendicular from point P on directrix and S is focus of the parabola, then the quadrilateral SPMT is a rhombus.
14. Reflection property of parabola : Any ray of light coming parallel to the axis of parabola, then after reflection through parabola, it passes through focus of the parabola and conversely, i.e., after the ray passes through focus, becomes parallel to axis of parabola after reflection through parabola.
11. A ray of light moving parallel to the x-axis gets reflected from a parabolic mirror whose equation is ()() 2 2 41yx−=+ . After reflection, the ray must pass through th e point P. Find P.
Sol. Reflected ray passes through the focus of the parabola ()() 2 2 41yx−=+ Thus, (2, 2) is the required point.
Try yourself:
11. A ray of light moving parallel to x-axis gets reflected from a parabolic mirror whose equation is () 2 4 – 0 xyy+= . After reflection the ray pass through the point(a, b). Then find the value of a + b. Ans: 2
TEST YOURSELF
1. The midpoint of chord 2x +y –5 = 0 of the parabola y2 = 4x is (1) (2, 1) (2) (1, 3)
(3) (3, –1) (4) 5 ,0 2
2. If x–2y–a =0 is a chord of y2 = 4ax, then its length is
(1) 45a (2) 5a (3) 20a (4) 40a
15. Diameter : The locus of midpoints of a system of parallel chords is called diameter. In parabola, diameter is a line parallel to principal axis. Equation of diameter of parabola 2 4 yax = corresponding to system of parallel chords , ymxc =+ is 2a y m =
3. If P is a point on the parabola y2 = 8x and A is the point (1,0), then the locus of the midpoint of the line segment AP is
(1) 2 1 4 2 yx =−
(2) y2 = 2(2x + 1)
(3) 2 1 2 yx=− (4) y2 = 2 x+1
4. The locus of midpoint of the line segment joining the focus to a moving point on the parabola y2 = 4ax is another parabola with directrix is (1) x = –a (2) 2 a x = (3) x = 0 (4) 2 a x =
CHAPTER REVIEW
Conic Section
1. The general equation of conic is 222220
of conic
0,0,,0 habe ∆≠=== A circle
2 0,0,1 abhe ∆≠−== A parabola
2 0,0,1 abhe ∆≠−>< An ellipse
2 0,0,1 abhe ∆≠−<> A hyperbola
2 0,0, abh ∆≠−< 0,2abe+== A rectangular hyperbola
2. Equation of Parabola
2 4 yax = 2 4 yax =−
Vertex (0, 0) (0, 0)
Focus (a, 0) (–a, 0)
Directrix 0 xa+= 0 xa−=
Axis 0 y = 0 y =
Latus
rectum 4a 4a
Focal distance
P(x,y) xa + ax
5. A ray of light along y = 4 is reflected at a point on y2 = 4x. The slope of reflected ray is
2 4 xay = 2 4 xay =−
Vertex (0, 0) (0, 0)
Focus (0, a) (0, –a)
Directrix 0 ya+= 0 ya−=
Axis 0 x = 0 x =
Latus rectum 4a 4a
Focal distance P(x,y) ya + ay
3. Equation of parabola whose axis is parallel to X axis is in the form of ()() 2 4;0ykaxha −=±−>
4. Equation of parabola whose axis is parallel to Y axis is in the form of ()() 2 4;0xhayka −=±−>
5. The equation of parabola having focus at () , αβ and directrix 0 lxmyn++= is ()()()2 22 22 lxmyn xylm ++ −α+−β= +
6. If 0 lxmyn++= , 0 mxlyk−+= are tangent at vertex, axis of the parabola then the equation of the parabola is 222 ()(4)|| mxlykalmlxmyn −+=+++
8. The equation of tangent to the parabola 0 S = at () 11 , Pxy is 1 0 S =
9. The equation of tangent to the parabola 2 4 yax = at () 11 , Pxy is ()11 2 yyaxx =+ .
10. The equation of the tangent to the parabola 2 4 yax = at t is 2 0 ytxat−−=
11. The condition for the line ymxc =+ to be a tangent to the parabola 2 4 yax = is a c m = and the point of contact is 2 2 , aa mm
.
12. The condition for the line 0 lxmyn++= to be a tangent to the parabola 2 4 yax = is 2 ln am = . Point of contact is 2 , nam ll
13. The condition for the line ymxc =+ to be a tangent to the parabola 2 4 xay = is 2 cam =− and the point of contact is () 2 2,amam
14. The condition for the line 0 lxmyn++= to be a tangent to the parabola 2 4 xay = is 2 n mal = . Point of contact is 2 , aln mm .
15. The condition for the line ymxc =+ to be a tangent to the parabola () 2 4 yaxa =+ is 1 cam m =+
16. Equation of common tangent to the parabolas 2 4 yax = , 2 4 xby = is () 11 2 33 3 0 axbyab++=
17. Locus of foot of perpendicular from focus on any tangent is tangent at the vertex of the parabola
18. The equation of normal to the parabola 2 4 yax = at () 11 , Pxy is ()() 111 20xxyyya −+−=
19. The condition for the line ymxc =+ to be normal to the parabola 2 4 yax = is that the slope form of the normal to the parabola 2 4 yax = is 3 2 ymxamam =−−
20. The condition for the line 0 lxmyn++= to be normal to the parabola 2 4 yax = is 32220alalmmn++=
21. The equation of normal to the parabola 2 4 yax = at () 2 ,2 tatat is 3 2 yxtatat +=+
22. The slope of normal to the parabola 2 4 yax = at t is t
23. If the normal to the parabola at 1t intersect the parabola again at 2 , t then 21 1 2 tt t =−−
24. If the normal to the parabola at t subtends a right angle at the focus, then 2 t =±
25. If the normal to the parabola at t subtends a right angle at the vertex, then 2 t =±
26. The length of the normal chord to the parabola 2 4 yax = at t is () 3 2 2 2 41at t +
27. For the parabola 2 4 yax = , the normal at 1t and the normal at 2 t intersect on the parabola, then 12 2 tt =
28. The point of intersection of normals drawn at 12 , tt to the parabola 2 4 yax = is (()() ) 2 2,12121212 aattttatttt ++−−+
29. Number of normal lines at () 11 , Pxy :
Substitute () 11 , Pxy in 3 2 yxtatat +=+ , we get () 3 11 20attaxy+−−= , it’s a cubic equation, it has at most three real roots and at least one real root, so that the number of normals to the parabola at any point is at least one and at most 3.
30. Number of normals to the parabola at () 11 , Pxy does not depends on the position of () 11 , Pxy , it depends on the number of distinct roots of the equation () 3 11 20attaxy+−−=
31. If 123 ,, ttt are the roots of the equation () 3 11 20attaxy+−−= , then 123 ,, ttt are called feet of the normals drawn from a point () 11 , Pxy
• 123 0 ttt++=
• 1 122331 2ax tttttt a ++=
• 1 123 ttty a =
32. If 123 ,, ttt are feet of the normals drawn from P to the parabola, then the circle passing through the points 123 ,, ttt also passing through vertex of the parabola.
33. Area of the triangle formed by feet of the normals 123 ,, ttt is a2 |(t1 – t2)(t2 – t3) (t3 – t1)|
34. Area of the triangle formed by three normals drawn to the parabola 2 4 yax = is () () ()()2 2 122331123 1 2 attttttttt =−−−++
35. The centroid of the triangle formed by the feet of the 3 normals to the parabola 2 4 yax = from a given point lies on the axis of the parabola.
36. If three normal drawn to parabola 2 4 yax = from a point (h,0) on the principal axis then 2 ha >
37. Reflection property of parabola: Any ray of light coming parallel to the axis of parabola then after reflection through parabola it passes through focus of the parabola and conversely. i.e., ray after passing through focus becomes parallel to axis of parabola after reflection through parabola.
38. Diameter: The locus of midpoints of system of parallel chords is called diameter, In parabola diameter is a line parallel to principal axis.
Pair of Tangents
39. If θ is acute angle between the pair of tangents drawn from () 11 , Pxy to the parabola 0 S = then
40. The locus of the point of intersection of perpendicular tangents is 0 xa+= , directrix of the parabola.
41. The angle between the tangents drawn from any point on the directrix to the parabola is 90° .
42. The angle between the tangents drawn at the end of the Latus rectum is 90° and point of intersection of these tangents is () ,0a
The point of intersection of tangents to the parabola 2 4at,12 yaxtt = is (() ) 1212 , attatt +
43. Let the tangents drawn to 2 4 yax = ()() 22 at,2,,21122 PatatQatat at T then
• The line joining T, mid point of PQ is parallel to axis.
• If SPQ is focal chord then STPQ ⊥ or focus S is foot of perpendicular of T on PQ.
Chord of Contact, Midpoint of Chord
44. Equation of diameter of parabola 2 4 yax = corresponding to system of parallel chords , ymxc =+ is 2a y m =
45. The equation of the chord of the parabola 2 4 yax = joining 12 , tt is () 12 12 22 yttxatt +=+
46. Focal chord: The condition for the chord joining 12 , tt to be focal chord is 12 1 tt =−
• The length of the focal chord passing through the point t is 2 1 at t
+
• If one end of focal chord is () 2 ,2 atat
then the other end is 2 2 , aa
• If () 11 , xy and () 22 , xy are ends of focal chord then 2 12 , xxa = and 2 12 4 yya =−
• If a focal chord makes an angle θ with x-axis then the length of the focal chord is 2 4csc a θ
• If PSQ is a focal chord and l is the length of semi Latus rectum of parabola 2 4 yax = then ,, SPlSQ are in HP, it means 112 SPSQl += or 111 SPSQa +=
JEE MAIN LEVEL
Level – I
Equation of Parabola
Single Option Correct MCQs
1. Axis of the parabola x2–3y–6x+6=0 is (1) x = –3 (2) y = –1 (3) x = 3 (4) y = 1
2. The ends of the Latus rectum of the parabola (x–2)2 = –6(y+1), are (1) (2, 7), (3, –7) (2) (0, 5), (0, –5) (3) (0, 7), (0, –5) (4) 55 5,,1, 22
3. The length of latus rectum of parabola 4y2+12 x–20y+67=0 is (1) 2 (2) 1 (3) 8 (4) 3
4. The length of Latus rectum of parabola x2–6x+16y+25=0 is (1) 5 (2) 4 (3) 16 (4) 5 3
5. The distance between the vertex and focus of the parabola x2–2x+3y–2=0 is (1) 4 5 (2) 3 4 (3) 1 2 (4) 5 6
6. The equation of directrix and Latus rectum of a parabola are 3 x–4y+27 = 0 and 3x–4y+2 = 0, then length of Latus rectum is (1) 5 (2) 10 (3) 15 (4) 20
7. The point (3,4) is the focus and 2 x –3 y +5 = 0 is the directrix of a parabola. Its latus rectum is (1) 2 13 (2) 4 13 (3) 1 13 (4) 3 13
8. The equation of the parabola with the focus (3,0) and the directrix x+3 = 0 is
(1) y2 = 3x (2) y2 = 6x (3) y2 = 12x (4) y2 = 2x
9. The vertex of parabola whose focus is (2,1) and directrix is x–2y+10 = 0 is (1) (2, 2) (2) (1, 3) (3) (3, 1) (4) (0, 5)
10. The focus and directrix of a parabola are (1, 2) and x+2y+9 = 0, then equation of Tangent at vertex is
(1) x+2y = 5 (2) x+2y = 2
(3) x+2y+5 = 0 (4) x+2y+2 = 0
11. The Latus rectum of the parabola 13[(x–3)2+(y–4)2] = (2x–3y+5)2 (1) 2
12. The focus of parabola whose vertex is (4, 5) and whose equation of directrix is x+y+1 = 0, is (1) (–9, 10) (2) (10, 9) (3) (9, 10) (4) (–10, –9)
13. If the vertex of a parabola is (4, 3) and its directrix is 3x+2y–7 = 0, then the equation of latusrectrum of the parabola is (1) 3x+2y–18 = 0 (2) 3x+2y–29 = 0
(3) 3x+2y–8 = 0 (4) 3x+2y–31 = 0
14. The parabola (y+1)2 = a(x–2) passes through the point (1, –2), then equation of its directrixes.
(1) 4x+1 = 0 (2) 4x–1 = 0
(3) 4x+9 = 0 (4) 4x–9 = 0
15. If L1 and L2 are the length of the segments of any focal chord of the parabola y2 = x, then 12 11 LL + is
(1) 2 (2) 3 (3) 4 (4) 10
Tangent and Normal to the Parabola
Single Option Correct MCQs
16. Equation of tangent to parabola y2 = 16x and perpendicular to the line x–4y–7 = 0
(1) 4x+y+1 = 0 (2) 4x+y+7 = 0
(3) 4x+y–1 = 0 (4) 4x+y–7 = 0
17. The line 4x+6y+9 = 0 touches the parabola y2 = 4x at the point
(1) 9 3, 4
(3) 9 , 3 4
(2) 9 3, 4
(4) 9 , 3 4
18. Equation of tangent to y2 = 8x at the end of the Latus rectum in 4th quadrant is
(1) x–y+2 = 0 (2) x–y+4 = 0
(3) x+y+2 = 0 (4) x+y+4 = 0
19. Point on the parabola y2 = 8x, the tangent at which makes an angle 4 π with axis is
(1) (2, 4) (2) (–2, 4) (3) (8, 8) (4) (–8, 8)
20. The slopes of tangents drawn from a point (4,10) to the parabola y2 = 9x are
21. The point of intersection of tangents at t1 and t2 to the parabola y2 = 12x is (1) [2t1t2, 2(t1 –t2)] (2) [3t1t2, 3(t1 –t2)] (3) [3t1t2, 3(t1+t2)] (4) [2t1t2, 2(t1+t2)]
22. If the tangents at t1 and t2 on y2 = 4ax meets on its axis then
38. The mid point of chord 2x+y–5 = 0 of the parabola y2 = 4x, is
(1) (2, 1) (2) (1, 3)
(3) (3, –1) (4) 5 ,0 2
39. The locus of point of intersection of perpendicular tangents drawn to x2 = 4ay is (1) x = 0 (2) y = –a
(3) x+a = 0 (4) y = a
40. The point of contact of the tangents drawn from the point (4, 6) to the parabola y2 = 8x
(1) (2, 4), (18, 12 (2) (2, 4) (8, 8)
(3) (8, 8) (18, 12) (4) (0, 0) (1, 22 )
41. The area of triangle formed by tangents and the chord of contact from(3, 4) to y2 = 2x, is
(1) 1010 (2) 210
(3) 1010 5 (4) 1010 3
Chord of Contact, Modpoint Chord
Single Option Correct MCQs
42. If the line x–2y+1 = 0 makes an intercept of length 230 on the parabola y2 = kx (k > 0) then k =
(1) 4 (2) 3
(3) 2 (4) 1
43. If 2x+y+k = 0 is a focal chord of y2+4x = 0 then k = (1) 2 (2) 4
(3) –4 (4) –2
44. The condition for the line 4x+3y+k = 0 to intersect y2 = 8x is
(1) 9 2 k < (2) k > 5
(3) 5 2 1 k < (4) k > 6
45. The equation of chord of the parabola y2 = 2x having (1, 1) as its midpoint is (1) x+y = 0 (2) x–y = 0
(3) x–y+1 = 0 (4) 2x–y = 0
46. If one end of focal chord of the parabola y 2 = 8 x is 1 ,2 2 , then the length of the focal chord is ______units
(1) 625 4 (2) 5 2
(3) 25 2 (4) 25
47. Length of chord of parabola y2 = 4ax whose equation is 2420yxa−+=
(1) 211a (2) 4 2a
(3) 8 2a (4) 6 3a
48. The equation of chord of the parabola y2 = 2x having (1,1) as its midpoint is
(1) x+y = 0 (2) x–y = 0
(3) x–y+1 = 0 (4) 2x–y = 0
49. The ordinate of a point on the parabola y2 = 18x is one third of its length of the Latus rectum. Then the length of subtangent at the point is
(1) 12 (2) 8 (3) 6 (4) 4
50. The mid point of chord 2x+y–5 = 0 of the parabola y2 = 4x is
(1) (2, 1) (2) (1, 3)
(3) (3, –1) (4) 5 ,0 2
51. Let O be the origin and A be a point on the curve y2 = 4x then locus of the midpoint of OA is
(1) x2 = 4y (2) x2 = 2y (3) y2 = 16x (4) y2 = 2x
52. If PSP 1 is a focal chord of a parabola y2 = 4ax and SL is its semi Latus rectum then SP, SL and SP1 are in (1) AP (2) GP (3) HP (4) AGP
53. On the parabola y 2 = 4 x the normal at (1, 2) meets again the parabola at (1) (–6, 9) (2) (–9, –6)
(3) (9, –6) (4) (–6, –9)
Level – II
Equation of Parabola
Single Option Correct MCQs
1. The length of latus rectum of the parabola whose focus is 22 sin2,cos2 22 uu gg αα
and directrix is 2 2 u y g = is
(1) 2 u g Cos α (2) 2 2 cos2 u g α
(3)
(4)
2. The equation of the parabola with focus (0,0) and directrix x+y = 4 is
(1) x2+y2–2xy+8x+8y–16 = 0
(2) x2+y2–2xy+8x+8y = 0
(3) x2+y2+8x+8y–16 = 0
(4) x2 –y2+8x+8y–16 = 0
3. A parabola with focus (1, 2) touches both the axes then equation of its directrix is (1) x+2y = 0 (2) 2x–y = 0
(3) 2x+y = 0 (4) x–2y = 0
4. The graph represented by the equations x = Sin2t and y = 2cost is
(1) a part of parabola
(2) a parabola
(3) a part of sine graph
(4) part of hyperbola
5. The curve described parametrically by x = t2+t+1 and y = t2 –t+1 represents (1) a hyperbola
(2) an ellipse
(3) a parabola
(4) a rectangular hyperbola
6. The vertex of the parabola whose parametric equations is x = t2 –t+1, y = t2+t+1, t ∈ R is (1) (1, 1) (2) (2, 2) (3) (3, 3) (4) 11 , 22
7. Let M be the foot of the perpendicular from a point P on the parabola y 2 = 8( x –3) on to its directrix and let S be the focus of the parabola. If ∆SPM is an equilateral triangle, then P =
(1) () 43,8 (2) () 8,43
(3) () 9,43 (4) () 43,9
8. An arch is in the shape of a parabola whose axis is vertically downwords and measures 80 mts across its bottom on the ground. Its highest point is 24 mts. The measure of the horizontal beam across its cross section at a height of 18 mts, is (1) 50 (2) 40 (3) 45 (4) 60
9. A telegraphic wire suspended between two poles of height 15 mts is in the shape of a parabola. The distance between the poles is 20 mts and maximum sag of the cable wire is 4 mt, then the height of the cable at a distance of 5 mt from one end is (1) 12 (2) 11 (3) 10 (4) 13
10. If the parabolas y2 = λx ; 25(x–3)2+(y+2)2 = (3x – 4y–2)2 are equal, then λ = (1) 8 (2) 6 (3) 4 (4) 5
11. Reflection of y2 = x about y-axis is (1) x+y2 = 0 (2) x2 –y = 0 (3) y2– 4x = 0 (4) x2+y = 0
Tangent and Normal to the Parabola
Single Option Correct MCQs
12. If the tangents drawn at the points P and Q on the parabola y 2 = 2 x –3 intersect at the point R (0, 1), then the orthocentre of the triangle PQR is (1) (0, 1) (2) (2, –1) (3) (6, 3) (4) (2, 1)
13. The radius of the circle which passes through the focus of parabola x2 = 4y and touches it at point (6,9) is 10 k . Then k = (1) 2 (2) 5 (3) 8 (4) 7
14. The equation of the common tangent to the parabola y2 = 32x and x2 = 256y, is (1) x+2y–32 = 0 (2) x+2y+32 = 0 (3) 2x+y–32 = 0 (4) 2x+y+22 = 0
15. The area of triangle formed by the points t1, t2 and t3 on y2 = 4ax is k|(t1 –t2)(t2 –t3)(t3 –t1)| then k = (1) 2 2 a (2) a2 (3) 2a2 (4) 4a2
16. The area of triangle formed by tangents at the parametric points t1, t2 and t3 on y2 = 4ax is k|(t1–t2)(t2–t3)(t3–t1)|, then k = (1) 2 2 a (2) a2 (3) 2a2 (4) 4a2
17. Given below are two statements: One is labbeled. Assertion (A) and the other is labelled (R)
Assertion (A): Orthocentre of the triangle formed by any three tangents to the parabola lies on the directrix of the parabola
Reason (R) : The orthocentre of the triangle formed by the tangents at t1, t2, t3 to the parabola y2 = 4ax is (–a, a(t1+t2+t3 +t1t2t3)).
In light of the above statements, choose the correct answer from the options given below.
(1) Both (A) and (R) are true and (R) is the correct explanation of (A)
(2) Both (A) and (R) are true and (R) is not the correct explanation of (A)
(3) (A) is true but (R) is false
(4) Both (A) and (R) are false
18. If the normals at P and Q meet again on y2 = 4ax at R, then centroid of ∆PQR lies on (1) axis (2) Latus rectum (3) directrix (4) parabola
19. The normal at a point 2 (,11 ) 2 btbt on a parabola meets the parabola again at the point () 2 22 , 2 btbt . Then (1) 21
(2) 21
(4)
20. On the parabola y2 = 4x the normal at (1,2) meets the parabola again at the point (1) (–6, 9) (2) (–9, –6) (3) (9, –6) (4) (–6, –9)
21. If (x1, y1), (x2, y2), (x3, y3) are feet of the three normals drawn from a point to the parabola y2 = 4ax, then 12 3 xx y ∑=
(1) 4a (2) 2a (3) 1 (4) 0
22. If the normals at P and Q meet again on the parabola y2 = 4ax, then the chord joining P and Q passes through a fixed point (1) (–a, 0) (2) (–2a, 0) (3) (–3a, 0) (4) (–4a, 0)
23. Normals are drawn from the point P with slopes m1, m2, m3 to y2 = 4x and, if locus of P such that m1.m2 = α is a part of the parabola itself, then α is equal to (1) 1 (2) 2 (3) 3 (4) –2
24. P and Q are two points on the parabola (y–2)2 = 4(x–3). The normals and tangents at P and Q form a square then point of intersection of tangents at P, Q is (1) (–1, 0) (2) (–4, 2) (3) (2, 2) (4) (3, 2)
25. If three distinct normals can be drawn to the parabola y 2–2 y = 4 x -9 from the point (2a, 0), then the range of value of a is
(1) no real values possible (2) (2, ∞ ) (3) (– ∞ , 2) (4) (–2, 2)
26. If two of the three feet of normals drawn from a point to the parabola y2 = 4x be (1,2) (1,-2), then third foot is (1) () 2,22 (2) () 2,22 (3) (0, 0) (4) (1, 1)
27. Point of concurrence of the normals drawn at (2, 8), (128, 64), and (162, –72) to the parabola y2 = 32x is (1) (2, 8) (2) (128, 64) (3) (162, –72) (4) (162, 72)
28. Given below are two statements: One is labelled Assertion (A) and the other is labelled Reason (R)
Assertion (A): If the line x = 3y+k touches the parabola 3y 2 = 4x, then k = 5
Reason (R) : Equation to the tangent y2 = 8x inclined at an angle 30° to the axis is x + 3 y+6 = 0
(1) Both (A) and (R) are true and (R) is the correct explanation of (A)
(2) Both (A) and (R) are true and (R) is not correct explanation of A)
(3) A is true but R is false
(4) Both A and R are false
29. Let x + y = k be a normal to the parabola y2 = 12 x. If P is length of the perpendicular from the focus of the parabola on to this normal. Then 4k–2P2 = (1) 1 (2) 0 (3) –1 (4) 2
30. If the normal to y 2 = 4 ax at t 1 cuts the parabola again at t2, then (1) 2 2 8 t ≤ (2) 2 2 8 t ≥
(3) 2 88 t −≤≤ (4)
2 8 t <
Pair of Tangents
Single Option Correct MCQs
31. A tangent and normal are drawn at P(2,-4) to the parabola and y 2 = 8 x , which meets directrix of parabola at A and B respectively. If Q ( a , b ) is a point such that AQBP is a square, then 2a+b equal to (1) –12 (2) –20 (3) –16 (4) –18
32. Two tangents are drawn from a point (–2, –1) to the curve y2 = 4x. If α is the angle between them, then | tan α | is equal to (1) 1 3 (2) 1 3 (3) 3 (4) 3
33. If the angle between the tangents drawn through the point (–2, –1) to the parabola y2 = 4x is θ then tan2 θ= (1) 3 (2) –3 (3) 3 4 (4) 3 4
34. (A) : The locus of the point of intersection of perpendicular tangents to y2 = 4ax is its directrix.
(R) : If θ is acute angle between the pair of tangents drawn from ( x 1 , y 1 ) to parabola S = 0, then 11 1 tan S xa θ = +
The correct answer is (1) Both A and R are true and R is the correct explanation of A
(2) Both A and R are true and R is not correct explanation of A
(3) Only A is true but R is not the correct explanation to A (4) Both A and R are false
35. If the tangent at the point P (2, 4) to the parabola y2 = 8x meets the parabola y2 = 8x+5 at Q and R, then the mid point of chord QR is
(1) (2, 4) (2) (4, 2) (3) (1, 3) (4) (3, 5)
36. If the common tangent to the circle x2+y2 = c2 and the parabola y2 = 4ax subtends an angle θ with x–axis then Tan2θ=
37. A chord is drawn through the focus of the parabola y2 = 6x such that its distance from the vertex of this parabola is 5 , 2 then its slope can be
(1) 5 2 (2) 3 2 (3) 2 5 (4) 2 3
38. An equilateral triangle is inscribed in the parabola y2 = 8x with one of its vertices is the vertex of the parabola. Then, the length of the side of that triangle is (1) 243 (2) 163 (3) 83 (4) 43
39. If an equilateral triangle is inscribed in a parabola y2 = 12x with one of the vertex is at the vertex of the parabola then its height is
(1) 243 (2) 163 (3) 36 (4) 24
40. If the line 330yx−+= cuts the parabola y2 = x+2 at A and B, then PAPB is equal to [where ()P3,0 = ]
(1) ()432 3 + (2) ()423 3 (3) 43 2 (4) ()232 3 +
41. If the point on the curve y2 = 6x, nearest to the point 3 3, 2
is ( α , β ), then 2( α + β ) is equal to ____.
(1) 4 (2) 8 (3) 6 (4) 9
42. Let L1 be the length of the common chord of the curves x2+y2 = 9 and y2 = 8x and L2 be the length of the latus rectum of y2 = 8x, then
(1) L1 > L2 (2) L1 = L2
(3) L1 < L2 (4) 1 2 L 2 L =
43. The locus of the mid points of the chords of the parabola x2 = 4py having slope m is a
(1) line parallel to x-axis at a distance |2pm| from it
(2) line parallel to y-axis at a distance |2pm| from it
(3) line parallel to y = mx, m ≠ 0 at a distance |2pm| from it
(4) circle with centre at the origin and radius |2pm|
44. An equilateral triangle is inscribed in parabola y 2 = 4 ax whose one vertex is at origin, then length of side of triangle
(1) 63a (2) 83a
(3) 43a (4) 23a
45. The locus of point of intersection of tangents drawn at the ends of chord of y 2 = 4 ax which subtends a right angle at vertex is
(1) x+a = 0
(2) x+2a = 0
(3) x+4a = 0
(4) y+a = 0
46. Let α1 and α2 be the ordinates of two points, A and B on a parabola y2 = 4ax and, let α3 be the ordinate of the point of intersection of its tangents at A and B. Then α3 –α2 =
(1) α3– α1
(2) α3 + α1
(3) α1
(4) α1 – α3
Level
– III
Single Option Correct MCQs
1. Length of the latus rectum of the parabola xya += (1) 2 a (2) 2 a (3) a (4) 2a
2. If A , B , C are 3 points on a parabola. ∆ 1, ∆ 2 are the areas of triangle formed by the points A, B, C and the tangents at A, B, C If ∆ 1, ∆ 2 are the roots of px2+qx+r = 0 then condition is
(1) 9q2 = 2pr (2) 9pr = 2q2
(3) 9p2 = 2qr (4) 2p2 = 9qr
3. If a tangent to the parabola y2 = 4ax meets the x-axis in T and the tangent at the Vertex A in P and the rectangle TAPQ is completed, then locus of Q is
4. Two parabola with a common vertex and with axes along x-axis and y-axis, respectively intersect each other in the first quadrant. If the length of latus rectum of each parabola is 3 then the equation of the common tangent to the two parabola is (1) 3(x+y) = 4 (2) 8(2x+y) = 3
(3) 4(x+y)+3 = 0 (4) x+2y+3 = 0
5. If a normal subtends a right angle at the vertex of a parabola y2 = 4ax, then its length is
(1) 23a (2) 43a
(3) 63a (4) 83a
6. If the two normals to y2 = 8x at (2, 4) and (18, 12) intersect at P(x1, y1) then foot of the 3 rd normal through P is (1) (32, 16) (2) (32, –16) (3) (–16, 32) (4) (2, 4)
7. The set of point on the axis of the parabola y 2–2 y –4 x +5 = 0 from which all the three normals to the parabola are real is:
(1) {(x, 1); x ≥3}
(2) {(x, –1); x ≥1}
(3) {(x,3); x ≥1}
(4) {(x, –3); x ≥3}
8. If the 4th term in the expansion of (px+ 1 x )n , n ∈ N is 5 2 and three normals to the parabola y2 = x are drawn through a point (q, 0), then (1) q = p (2) q > p (3) q < p (4) pq = 1
9. If two different tangents of y2 = 4x are the normals to x2 = 4by, then (1) 1 22 b > (2)
14. The coordinates of a point R on the axis of the parabola y 2 = 4 ax such that if PQ is a variable chord through R, then 22 11 PRQR + is independent of the slope of the chord is
(1) (2a, 0) (2) (–2a, 0) (3) (2a, 0) (4) (2a, 0)
15. If reflection of the parabola y2 = 4(x–1) in the line x+y = 2 is the curve Ax+By = x2, then the value of (A+B) is (1) 0 (2) 1 (3) 2 (4) 3
16. The locus of middle points of normal chords of the parabola y2 = 4ax is
(1) 2 3 2 4 2 2 yaxa ay +=− (2)
10. If two tangents drawn through the point (α, β) to the parabola y2 = 4x such that slope of one tangent is 3 times of the other then (1) 2 2 9 βα = (2) 2 3 16 αβ = (3) 2α = 9β2 (4) 2β = 9α2
11. Two tangents are drawn from a point to y2 = 4ax, if these are normals to x2 = 4by then (1) a2 > b2 (2) a2 > 4b2 (3) a2 > 8b2 (4) 4a2 < b2
12. Number of focal chords of the parabola y2 = 9x whose length is less than 9 is (1) 2 (2) 5 (3) 1 (4) 0
13. A circle of radius 4 , drawn on a chord of the parabola y2 = 8x as diameter, touches the axis of the parabola. Then, the slope of the chord is (1) 1 2 (2) 3 4 (3) 1 (4) 2
17. Let P be a variable point on the parabola y = 4x2+1. Then, the locus of the midpoint of the point P and the foot of the perpendicular drawn from the point P to the line y = x is
(1) 2(3x–y)2+(x–3y)+2 = 0
(2) (3x–y)2+2(x–3y)+2 = 0
(3) (3x–y)2+(x–3y)+2 = 0
(4) 2(x–3y)2+(3x–3y)+2 = 0
18. A ray of light moving parallel to x-axis gets reflected from a parabolic mirror( y –2) 2 = 4( x +1). After reflection the ray must pass through
(1) (0, 2) (2) (0, –2)
(3) (2, 0) (4) (1, 2)
Numerical Questions
19. If a parabolic reflector is 20 cm in diameter and 5 cm deep, then the distance from vertex to focus is _____.
20. A parabola with vertex (2,3) and axis parallel to the y-axis passes through(4,5). Then its length of latus rectum is ______.
21. If ( x 1, y 1), ( x 2, y 2) are the extremities of a
THEORY-BASED QUESTIONS
Very Short Answer Questions
1. What is the name of the conic, whose eccentricity is more than 1?
2. What is the shortest focal chord of parabola?
3. What is the least length of the focal chord?
4. What is the equation to the locus of point of intersection of perpendicular tangent s drawn to the parabola 2 4 yax = ?
5. What is the axis of the symmetry of the parabola 2 yaxbxc =++ ?
6. What is the maximum number of common normals that can be drawn to the two parabolas 224,4 yaxxby == ?
7. If λ is the area of triangle formed by three points P,Q,R on the parabola 2 4 yax = and µ is the area of the triangle formed by the points of intersection of tangents drawn at these points P,Q,R. What is the value of λ µ ?
8. What is the length of double ordinate of the parabola 2 4 yax = which subtends a right angle at the vertex?
9. A parabola 2 yaxbxc =++ crosses X- axis at two points at ()() ,0,,0αβ both to the right of the origin. A circle passes through these two points . what is the length of a tangent from the origin to the circle?
focal chord of the parabola y 2 = 16 x, then 4x1x2+y1 y2 = ________.
22. If the line x –1 = 0 is the directrix of the parabola y2 –kx+8 = 0, k≠0 and the parabola intersects the circle x 2+ y 2 = 4 in two real distinct points, then the value of |k| is _____.
23. Radius of the largest circle which passes through the focus of the parabola y 2 = 4 x and contained in it, is _____.
10. What is the locus of point of trisection of the double ordinates of the parabola 2 4 yax = ?
Statement-based Questions
Each question has two statements, statement I and statement II. In light of the given statements, choose the most appropriate answer from the options given below.
(1) Both statement I and statement II are correct.
(2) Both statement I and statement II are incorrect.
(3) Statement I correct but statement II is incorrect.
(4) Statement I incorrect but statement II is correct.
11. S - I : If PSQ is focal chord of the parabola 2 4 yax = then 111 SPSQa +=
S - II : If 12 , ll area lengths of segments of a focal chord of the parabola 2 4 yax = then length of the latus rectum of the parabola is 12 12 ll ll +
12. S - I : The equation of common tangent to circle 222 2 xya += and a parabola 2 8 yax = is () 2 yxa =±+
S - II : Equation of common tangent to the parabola 2 4 yx = and 2 4 xy = is 20 xy++=
13. S - I : The length of normal chord of the parabola 2 4 yax = which subtends a right angle at the vertex is 63a
S - II : The length of normal chord of the parabola 2 4 yax = which makes an angle 45° with the axis of the parabola is 82a
14. S - I : The condition for the line ymxc =+ is a tangent to the parabola 2 4 xay = is a c m =
S - II : If the segment intercepted by the parabola 2 4 yax = with the line 0 lxmyn++= which subtends an angle 90° at the vertex then 40 aln+=
15. S - I : The equation of axis of the parabola
()() {}() 222 2553341 xyxy −+−=−+ is 3410 xy−+=
S - II : The equation of latus rectum of the parabola
()() {}() 222 2553341 xyxy −+−=−+ is 3430 xy−−=
16. S - I : The equation of directrix of the parabola
()() {}() 22 2 1691351217 xyxy −+−=−+ is 512170 xy−+=
S - II : The length of latus rectum of the parabola
()() {}() 22 2 1691351217 xyxy −+−=−+ is 28 13
17. S - I : If two tangents drawn from the point () , αβ to the parabola 2 4 yx = be such that the slope of one tangent is double the other then 2 29βα =
S - II : The length of focal chord of the parabola 2 4 yax = which is inclined at 30° to the axis of the parabola is 16a .
18. S - I : The circle 22 20, xyxR λλ ++=∈ touches the parabola 2 4 yx = externally if 0 λ >
S - II : A variable circle passes through a fixed point () ,0a and touches a fixed line 0 x = then locus of its centre is a parabola.
19. S - I : The area of the triangle inscribed in the parabola 2 4 yx = with vertices whose ordinates are 1,2,4 is 3 4 sq.units
S - II : The area of triangle formed by the lines joining the vertex of the parabola 2 12 xy = to the ends of its latus rectum is 18 sq units
Assertion Reason Questions
In each of the following questions, a statement of Assertion (A) is given, followed by a corresponding statement of Reason (R). Mark the correct answer as
(1) if both (A) and (R) are true and (R) is the correct explanation of (A)
(2) if both (A) and (R) are true but (R) is not the correct explanation of (A)
(3) if (A) is true but (R) is false
(4) if both (A) and (R) are false
20. (A) : The length of the side of an equilateral triangle inscribed in the parabola 2 4 yax = whose one verted is at (0,0) is 83a
(R) : The length of the double ordinate of the parabola 2 4 yax = which subtend an angle θ at the vertex is 8 cot 2 a θ .
21. (A) : The curve represented by the equation 22 924162015600 xxyyxy −+−−−= is a parabola
(R) : A general second degree equation 222220 axhxybygxfyc +++++= represents a parabola if abc + 2fgh –af2 – bg2 – ch2 ≠ 0, h2 < ab, a ≠ b
22. (A) : If normals to the curve 2 yx = at the points P,Q and R passis through the 3 0, 2 then the circum radius of triangle PQR is 1 .
(R) : The centroid of the triangle formed by the feet of the 3 normals to the parabola 2 4 yax = from a given point lies on the axis of the parabola
23. (A) : The latus rectum is the shortest focal chord of a parabola.
(R) : As the length of a focal chord of the parabola 2 4 yax = is 2 1 at t + , it is minimum at 1 t =
24. (A) : The area of triangle formed by pair of tangents drawn from a point () 12,8 to the parabola 2 4 yx = and their chord of contact is 32 sq units.
(R) : Area of the triangle formed by the tangents from () 11 , Pxy and chord of contact is () 3 2 11 2 S a
25. (A) : If the line 10 x −= is the directrix of the parabola 2 8 ykx=− the 4 k =
(R) : Equation of directrix of the parabola ()() 2 4 yaxβα−=− is 0 xa α −+=
26. (A) : If the tangents drawn to the parabola 2 4 yax = at two points ()() 1122 ,,, PxyQxy intersect at () 33 , Txy then 132 ,, xxx are in GP.
JEE ADVANCED LEVEL
Multiple Option Correct MCQs
1. If l, m are variable real numbers such that 5 l 2 +6 m 2 –4 lm +3 l = 0 then variable line lx+my = 1 always touches a fixed parabola whose axis is parallel to x-axis
(R) : If the tangents drawn to the parabola 2 4 yax = at two points ()() 1122 ,,, PxyQxy intersect at () 33 , Txy then 132 ,, yyy are in AP.
27. (A) : The tangents and normal drawn to the parabola 2 4 yax = at the ends of latus rectum form a square of area 2 8a
(R) : The tangents drawn to the parabola 2 4 yax = at the ends of latus rectum intersect at () ,0a . Point of intersection of normal drawn to 2 4 yax = at the ends of latus rectum intersect at () 3,0 a
28. (A) : If P(-3,2) is one end of the focal chord PQ of the parabola 2 440yxy++= then the slope of the normal at Q is 1 2
(R) : The tangent at one end of the focal chord is parallel to the normal at the other end of the focal chord.
29. (A) : The conic represented by 1 xy ab += is a parabola.
(R) : 222220Saxhxybygxfyc ≡+++++= represents a parabola if
i. 222 20abcfghafbgch+−−−≠
ii. 2 hab =
(1) vertex of the parabola is 54 , 33
(2) focus of the Parabola is 1 4 , 33
(3) directrix of the parabola is 3 x+11 = 0
(4) directrix of the parabola is x+1 = 0
2. A parabola touches the lines y = x and y = -x at P (3, 3) and Q (2,-2) respectively. Then
(1) Focus is 306 , 1313
(2) Equation of directrix is x+5y = 0
(3) Equation of line through origin and focus is x+5y = 0
(4) Equation of line through origin and parallel to axis is x–5y = 0
3. If P 1 P 2 and Q 1 Q 2 , two focal chords of a parabola are at right angles, then:
(1) area of the quadrilateral P 1Q 1P 2Q 2 is minimum when the chords are inclined at an angle 4 π to the axis of the parabola
(2) minimum area is twice the area of the square on the latus rectum of the parabola
(3) minimum area of P1Q1P2Q2 cannot be found
(4) minimum area is thrice the area of the square on the latus rectum of the parabola
4. Two tangents on a parabola are x–y–1 = 0, x+y+11 = 0, if S(1, 3) is focus of the parabola. P, Q are ends of focal chord then
(1) Equation of tangent at vertex is 3y–2x+1 = 0
(2) Equation of directrix is 3 y–2x+8 = 0
(3) Length of latus rectum 30 13 =
(4) 11213 15 SPSQ +=
5. The equation of a parabola is 25[( x –2) 2 +( y +5) 2 ] = (3 x +4 y –1) 2 for this parabola.
(1) tangent at the vertex is 6 x + 8y + 13 = 0
(2) focus = (2, -5)
(3) directrix has the equation 3x + 4y – 1 = 0
(4) axis has the equation 4 x - 3y - 23 = 0
6. P is a point which moves in the xy plane such that the point P is nearer to the centre of a square than any of the sides. The four vertices of the square are (±a, ±a). The region in which P will move is bounded by parts of parabolas of which one has the equation
(1) y2 = a2+2ax
(2) x2 = a2+2ay
(3) y2 +2ax = a2
(4) x2+ 2ay = a2
7. In the figure a parabola is drawn to pass through the vertices B, C and D of the square ABCD. If A(2,1) C(2, 3). Then
(1) Vertex is (2, 3) (2) Focus is 11 2, 4
(3) B(3, 2) (4) D(1, 2)
8. P is a point on the parabola y2 = 4x and Q is a point on the line 2x + y + 4 = 0. If the line x – y + 1 = 0 is the perpendicular bisector of PQ, then the coordinates of P can be:
(1) (1, –2) (2) (4, 4)
(3) (9, –6) (4) (16, 8)
9. The tangent PT and the normal PN to the parabola y2 = 4ax at a point P on it meet its axis at points T and N , respectively. The locus of the Centriod of the triangle PTN is a parabola whose
(1) vertex is 2 ,0 3 a
(2) directrix is x = 0
(3) latus rectum is 2 3 a
(4) focus is (a, 0)
10. The value of ‘a’ for which two curves
y = ax2+ax+ 1 24 and x = ay2+ay+ 1 24 touch each other is
(1) 2 3 (2) 1 3 (3) 1 2 (4) 3 2
11. Consider a parabola with vertex A. Let P, Q be two points on it. The equations of tangents at P, Q, A to the parabola are 3x+4y–7 = 0,2 x+3y–10 = 0 and x–y = 0 respectively, then which is/are correct?
(1) vertex of the parabola is (5,5) (2) focus of the parabola is (4, 5)
(3) length of Latus rectum of the parabola is 22
(4) axis of the parabola is x+y+9 = 0
12. The equations of the common tangents to the parabola y = x2 and y = –(x – 2)2 are
(1) y = 4(x – 1) (2) y = 0
(3) y = –4(x – 1) (4) y = 30x – 50
13. Tangents are drawn from (-2, 0) to y2 = 8x, radius of circle(s) that would touch these tangents and the corresponding chord of contact, can be equal to,
(1) ()421 + (2) ()421
(3) 82 (4) 42
14. Let L be a normal to the parabola y2 = 4ax. If L passes through the point (9, 6), then L is given by
(1) y–x+3 = 0 (2) y+3x–33 = 0
(3) y+x–15 = 0 (4) y–2x+12 = 0
15. For the parabola 1 23 xy+= , which touches x-axis at (2, 0) and y-axis at (0, 3) which of the following is/are true?
(1) Equation of axis is 39 x – 26y – 30 = 0
(2) Focus is 1812 , 1313
(3) Directrix is 2x + 3y = 0
(4) Latus rectum is ()3/2 144 13
16. The focus of a parabola is S(–1, 1) and the equation of the directrix is 4x +3 y –24 = 0 then
(1) Equation of its axis is 3 x–4y+7 = 0
(2) Coordinates of its vertex is 5 1, 2
(3) Equation of the tangent at vertex is 4x+3y–6 = 0
(4) Length of latus rectum is 10
17. Consider the parabola represented by the parametric equations x = t2-2t+2 ; y = t2+2 t+2. Then which of the following is/are true?
(1) length of latus rectum is 42
(2) vertex of the parabola is (2, 2)
(3) x+y = 4 is tangent at vertex
(4) focus of the parabola is (3, 3)
18. The parabola y = x2+px+q cuts the straight line y = 2x–3 at a point with abscissa 1. If the distance between the vertex of parabola and the x-axis is least, then
(1) p = 0 and q = –2
(2) p = –2 and q = 0
(3) Least distance between vertex of the parabola and x-axis is 2
(4) Least distance between vertex of the parabola and x-axis is 1
be 3 points on the parabola y 2 = 4 ax . If the orthocenter of ∆ ABC is focus S of the parabola then (none of the vertices of the triangle is vertex of the parabola)
(1) t1t2+t3t2+t3t1 = –5
(2) 122331 111 1 tttttt ++=−
(3) If t1 = 1 then t2+t3 = 0
(4) (1+t1)(1+t2)(1+t3) = –4
Numerical/Integer Value Questions
20. All the three vertices of an equilateral triangle lie on the parabola y = x 2, and one of its sides has a slope of 2. The x-coordinates of the three vertices have a sum equal to p q where p and q are relatively prime positive integers, then the value of ( q – p)is _____.
21. If y = x+k is normal to the parabola x2 = 8y then value of k is ______.
22. The mirror images of y2 = 4x in the tangent at the point (1, 2) is (x+1)2 = 2k(y–1) then k = _______
23. From the point (3,0) three normals are drawn to the parabola y2 = 4x. Feet of these normals are P, Q, R then the area of triangle PQR is _____.
24. If parabolas y2 = ax and 25[(x–3)2+(y+2)2] = (3x–4y–2)2 are equal then the value of a is _____.
25. The number of common tangents for the circle x2+y2–4x+3 = 0 and parabola y2 = 2x is _____.
26. The length of the shortest path that begins at the point (-1,1), touches the x -axis and then ends at a point on the parabola ( x – y ) 2 = 2( x + y –4), is a 2 (where a ∈ N ); then the value of a is _____.
27. If the length of the latus rectum of the parabola 169{(x–1)2+(y–3)2}=(5x–12y+17)2 is L, then the value of 13 4 L is ____.
28. The equation of the line touching both the parabolas y2 = 4x and x2 =–32y is ax+by+c = 0. Then the value of a+b+c _____.
29. The radius of the circle which passes through the focus of the parabola x 2 = 4 y and touches it at the point (6, 9) is 10 k then k = _____.
30. A tangent at the point P(20, 25) to a parabola intersects its directrix at(–10, 9). If the focus of the parabola lies on x–axis, the number of such parabolas are
31. Let x + y = k be a normal to the parabola y2 = 12x. If p is length of the perpendicular from the focus of the parabola on to this normal, then 4k–2p2 = ______.
32. If a and b are the segments of a focal chord and 2c the latus rectum of a parabola and 33 3 ab K c + > then K = _____.
33. P is a point on the parabola y 2 = 3(2 x –3) and M is the foot of perpendicular drawn from P on the directrix of the parabola then the length of each side of the equilateral triangle SMP, when S is focus of the parabola is _____.
34. Consider a parabola 2 2 yx = . From a point where its directrix intersects the line 22230 xy−+= , a normal to the parabola is drawn, meeting the parabola at M (foot of normal) and N. R is point of intersection of tangent at M and chord ON produced (O is origin). If angle ∠ MRN is θ then value of |tan θ | is _______
Passage-based Questions (Q. 35–37)
P1: y2 = 4ax, P2 : y2 =-4ax, L: y = x
35. Area of the rectangle formed by joining the vertices of the latusrecta of the two parabolas P1 and P2 is
(1) 2a2 sq.units (2) 4a2 sq.units
(3) 8a2 sq.units (4) 16a2 sq.units
36. Equation of the tangent at the point on the parabola P1 where the line L meets the parabola is
(1) x-2y+4a = 0 (2) x+2y–4a = 0
(3) x+2y–8a = 0 (4) x–2y+8a = 0
37. The coordinates of other extremity of a focal chord of the parabola P2, one of whose extremity is the point of intersection of L and P2 is
(1) (–a, 2a) (2) (–a /4, a)
(3) (–a /4, –a) (4) (–a, –2a)
(Q. 38–40)
A parabola whose focus is S(3,4) is touching the coordinate axes
38. The equation of the circle whose diameter is the portion of tangent at vertex of the parabola between the coordinate axis is
(1) x2+y2–3x–4y = 0 (2) x2+y2+6x+8y = 0
(3) x2+y2–6x–8y = 0 (4) x2+y2+3x+4y = 0
39. The equation of axis of the parabola is
(1) 4x–3y+7 = 0 (2) 3x–4y = 0
(3) 4x–3y = 0 (4) 3x–4y+7 = 0
40. If P, Q are ends of focal chord of the parabola then 11 SPSQ +=
(1) 12 5 (2) 5 12 (3) 6 5 (4) 5 6
(Q. 41–43)
y2 = 4x and y2 = –8(x– a) intersect at points A and C . Points O (0, 0), A , B ( a , 0), C are concyclic.
41. The length of common chord of parabolas is
(1) 26 (2) 43 (3) 65 (4) 82
42. The area of cyclic quadrilateral OABC is (1) 243 (2) 482 (3) 126 (4) 185
43. Tangents to parabola y 2 = 4 x at A and C intersect at point D and tangents to parabola y2 = – 8(x–a) intersect at point E, then the area of quadrilateral DAEC is
(1) 962 (2) 483 (3) 545 (4) 366
(Q. 44–45)
Consider the parabola ( y –2) 2 = 4( x –2) and (x–2)2 = 4(y–2). Let S = 0 be the largest circle touching the two parabola’s internally and S’ = 0 be the circle described on the common chord of the two parabolas as diameter. Then answer the following questions.
44. Centre of S = 0 is (1) () 22,22 (2) (2, 2) (3) (4, 4) (4) () 23,23
45. If x 2+ y 2 = k cuts S ’ = 0 orthogonally then k = (1) 24 (2) 12 (3) 8 (4) 4
46. Radius of S = 0 is (1) ()2108 (2) ()2108 +
(3) () 108 + (4) () 108
(Q. 47–48)
Consider one side AB of a square ABCD, (read in order) on the line y = 2x–17, and the other two vertices C, D on the parabola y = x2 .
47. Minimum intercept of the line CD on y-axis, is (1) 3 (2) 4 (3) 2 (4) 6
48. Maximum possible area of the square ABCD can be (1) 980 (2) 1160 (3) 1280 (4) 1520
(Q. 49–50)
A right angle triangle ABC is inscribed in a parabola y2 = 4x, where A is vertex of parabola and ∠ BAC = 2 π . If AB = 5
49. If B(t2, 2t); t >0 lies in first quadrant then slope of the side AB is ______
50. Area of the triangle ABC is k then 10 k = ______.
(Q. 51–53)
If P is a point moving on a parabola y2 = 4ax and Q is point moving on the circle x 2+ y 2 –24ay+128a2 = 0. The points P and Q will be closest when they lie along a normal to the parabola y2 = 4ax passing through the centre of the circle.
51. If the normal at ( at 2, 2 at ) of the parabola passes through the centre of the circle then the value of t must be (1) 1 (2) 2 (3) 3 (4) 4
52. When P and Q are closest, then P must be the point
(1) (a, 2a) (2) () 2,22aa
(3) (4a, 4a) (4) (a, a)
53. The shortest distance between P and Q must be
(1) () 21 a (2) ()21 5 a
(3) ()41 5 a (4) ()41 5 a +
(Q. 54–56)
A tangent is drawn at any point P(t) on the parabola y 2 = 8 x and on it is taken a point Q ( α , β ) from which pair of tangents QA and QB are drawn to the circle x 2 + y 2 = 8. Using this information answer the following questions.
54. The locus of the point of concurrency of the chord of contact AB of the circle x2 + y2 = 8 is
(1) y2 – 2x = 0 (2) y2 – x2 = 4
(3) y2 + 4x = 0 (4) y2 – 2x2 = 4
55. The point from which perpendicular tangents can be drawn both to the given circle and the parabola is
(1) () 4,3 ± (2) () 1,2
(3) () 2,2 (4) () 2,23−±
56. The locus of circumcentre of ∆ AQB, if t = 2 is
(1) x – 2y+ 2=0 (2) x + 2y – 4 = 0
(3) x – 2y + 4 = 0 (4) x + 2y + 4 = 0
(Q. 57–58)
The chord AC of the parabola y2 = 4ax subtends an angle of 90° at points B and D on the parabola. If A , B , C, and D are represented by t1, t2, t3 and t4 then
57. Value of 24 13 2 tt tt + = + _______.
58. The y -coordinate of the midpoint of the points of intersection of the tangents at A, C and B, D is _______.
(Q. 59–60)
The normal at any point (x 1, y 1) of curve is a line perpendicular to tangent at the point ( x 1 , y 1 ). In case of parabola y 2 = 4 ax , the equation of normal is y = mx –2am –am 3 ( m is slope of normal). The shortest distance between any two curves always exist along the common normal.
59. The shortest distance between the parabolas 2 y2 = 2x–1, 2x2 = 2y–1 is p, then [p2] (where step denotes greatest integer function) = _____.
60. Number of normals drawn from ( 7 6 , 4) to parabola y2 = 2x–1 is _____.
Matrix Matching
61. Match the following list-I with list-II List - I List - II
(A) Locus of point of intersection of perpendicular tangents to y2+4ax = 0 is (P) x = a
(B) Locus of P whose chord of contact subtends a right angle at vertex of y2+4 ax = 0 is (q) x = 4a
(C) Common normal to y2+4 ax = 0 and x2 = 4ay (r) x + 3a = 0
(D) Locus of P, if tangents from P to the parabola y2 = 4ax intersect coordinate axis in concyclic points (s) x – y = 3a
(1) A-p, B-q, C-s, D-p
(2) A- r, B-p, C-q, D-s
(3) A-r, B-s, C-q, D-p
(4) A-p, B-q, C-r, D-s
62. Match the items of List-I with the items of List-II and choose the correct option.
List - I List - II
(A) The x-coordinate of points on the axis of the Parabola y2 - 4x - 2y + 5 = 0 from which all the three normals to the parabola are real is (p) 4
(B) The x-coordinate of points on the axis of the parabola 4y2 –32x + 4y + 65 = 0 from which all the three normals to the parabola are real is (q) 5
(C) The x-coordinate of points on the axis of the Parabola 4y2 –16x – 4y + 41 = 0 from which all the three normals to the parabola are real is (r) 6
(D) The x-coordinate of focus of the Parabola y2 – 4y – 8x + 28 = 0 (s) 7
(1) A-ps, B-s,C-pqrs, D-q
(2) A-q, B-s, C-qrs, D-pqrs
(3) A-pqrs, B-s, C-qrs, D-q
(4) A-pqrs, B-q, C-s, D-q
63. y 2 = 12 x is a parabola and P(9, μ) be a point in the plane then match the following
List - I
List - II
(A) Only one normal of y2 = 12x will go through P (p) 2 || 3 µ=
(B) No normal of y2 = 12x will go through P (q) 2 || 3 µ<
(C) Exactly two normals will go through P (r) null set
(D) Exactly three distinct normals will go through P (s) 2 || 3 µ>
(1) A-r, B-s, C-p, D-q
(2) A-s, B-p, C-r, D-q
(3) A-s, B-r, C-q, D-p
(4) A-s, B-r, C-p, D-q
64. Match the following list-I with list-II
List - I List - II
(A) Radius of the largest circle which passes through the focus of the parabola y2 = 4x and contained in it, is (p) 16
(B) Two perpendicular tangents PA and PB are drawn to the parabola y2 = 16x then length AB may be (q) 5
(C) The shortest distance between parabolas y2 = 4x and y2 = 2x – 6 is d then d2 (r) 8
(D) The harmonic mean of the segments of a focal chord of Parabola y2 = 8x (s) 4
(1) A-p, B-q,C-s, D-s
(2) A-s, B-p, C-q, D-s
(3) A-p, B-s, C-q, D-s
(4) A-s, B-q, C-p, D-s
65. Normals are drawn from point (4, 1) to the parabola y 2 = 4 x . The tangents at the feet of normals to the parabola y2 = 4x from a triangle ABC
List - I
List - II
(A) The distance of focus of parabola y2 = 4x from centroid of ∆ABC is (p) 5 3
(B) The distance of focus of parabola y2 = 4x from orthocentre of ∆ABC is (q) 10 2
(C) The distance of focus of parabola y2 = 4x from circumcentre of ∆ABC is (r) 7 2
(D) Area of ∆ABC is (s) 5 2 (t) 5
(1) A-p, B-t,C-q, D-s
(2) A-s, B-t, C-q, D-p
(3) A-p, B-r, C-q, D-t
(4) A-p, B-s, C-q, D-r
66. AB is a chord of the parabola y2 = 4x such that the normals at A and B intersect at the point C(9, 6).
List - I
List - II
(A) The length AB (p) 20
(B) The area of ∆ABC (q) 4 13
(C) The distance of the origin from the line through AB (r) 13
(D) The area bounded by the coordinate axes and the line through AB (s) 4 3
(A) Locus of point of intersection of perpendicular tangents is (p) 12x–5y–2 = 0
(B) Locus of foot of perpendicular from focus upon any tangent is (q) 5x+12y–29 = 0
(C) Line along which minimum length of focal chord occurs (r) 12x–5y+3 = 0
(D) Line about which parabola is symmetric is (s) 24x–10y+1 = 0
(1) A-r, B-q, C-p, D-s
(2) A-p, B-s, C-r, D-q
(3) A-q, B-s, C-p, D-r
(4) A-r, B-s, C-p, D-q
BRAIN TEASERS
1. If the normal from any point to the parabola x2 = 4y cuts the line y = 2 in points whose abscissa are in AP , then the slopes of the tangents at the three conormal points are in (1) AP (2) GP (3) HP (4) none of these
2. Consider a parabola 2 4 x y = and the point
F(0, 1). Let A1(x1, y1), A2(x2, y2), .....An(xn, yn), are n points on the parabola such that xk > 0 and ∠ OFAk= 2 k n π (k=1,2,3,...., n) Then the value of 1 1 lim n k nk FA n →∞ = ∑ is (1) 2 π (2) 4 π (3) 2 π (4) 4 π
3. If the normal to a parabola y 2 = 4 ax at P meet the curve again in Q and if PQ and the normal at Q makes angle α and β respectively with the x-axis, then {–tan α (tan α +tan β )}
(1) 1 3 (2) 3 (3) –2 (4) 2
4. If the normals at three points P, Q, R of the parabola y2 = 4ax meet at a point O and S be its focus, then |SP| . |SQ| .|SR| is equal to
FLASHBACK (P revious JEE Q uestions )
JEE Main
1. If the shortest distance of the parabola y2 = 4x from the centre of the circle x2 + y2 – 4x – 16y + 64 = 0 is d, then d2 is equal to (27th Jan 2024 Shift 1) (1) 16 (2) 24 (3) 20 (4) 36
(1) a2 (2) a(SO)3 (3) a(SO)2 (4) none of these
5. min [( x 1 – x 2 ) 2 +(3+ 2 1412 xx ) 2 ], x 1 , x2∈ R, is (1) 451 + (2) 322 (3) 51 + (4) 51
6. If y = x +1 is axis of parabola, x + y = 4 is tangent of same parabola at its vertex and y = 2x+3 is one of its tangents, then the length of the latus rectum of parabola is 2 a b where a and b are relatively prime natural numbers, then 1 12 ab++ = ______.
7. The mirror image of any point on the directrix of the parabola y2 = 4(x+1)in the line mirror x+2y–3 = 0 lies on 3x–4y+4k = 0 then k = _____.
8. Let L1 = x+y = 0 L2 = x–y = 0 are tangents to a parabola whose focus S(1, 2). If the length of latus rectum of the parabola can be expressed as m n (where m,n are coprime) then the value of 11 mn + is _____.
2. Let R be the focus of the parabola y2 = 20x and the line y = mx+c intersect the parabola at two points P and Q. Let the point G(10, 10) be the centroid of the triangle PQR. If c–m = 6, then (PQ)2 is(8th Apr 2023 Shift 1) (1) 317 (2) 325 (3) 346 (4) 296
3. Let A (0,1), B (1,1) and C (1,0) be the mid points of the sides of a triangle with incentre at the point D. If the focus of the parabola y2 = 4ax, passing through D is () 2, 0 αβ + , where α and β are rational numbers, then 2 α β is equal to (8th Apr 2023 Shift 2) (1) 12 (2) 9 2 (3) 8 (4) 6
4. Let PQ be a focal chord of the parabola y2 = 36x of length 100, making an acute angle with the positive x -axis. Let the ordinate of P be positive and M be the point on the line segment PQ such that PM:MQ = 3: 1. Then which of the following points does NOT lie on the line passing through M and perpendicular to the line PQ?
(13th Apr 2023 Shift 1)
(1) (–6, 45) (2) (3, 33) (3) (6, 29) (4) (–3, 43)
5. The equations of the sides AB and AC a triangle ABC are ( λ +1) x + λ y = 4 and λ x+(1– λ )y+ λ = 0 respectively. Its vertex A is on the y-axis and its orthocentre is (1,2). The length of the tangent from the point C to the part of the parabola y2 = 6x in the first quadrant is (24th Jan 2023 Shift 1) (1) 2 (2) 6 (3) 22 (4) 4
6. The distance of the point () 6,22 from the common tangent y = mx+c, m>0, of the curves x=2y2 and x =1+y2 is (25th Jan 2023 Shift 1)
(1) 53 (2) 14 3 (3) 1 3 (4) 5
7. The equation of two sides of a variable triangle are x = 0 and y = 3 and its third side is a tangent to the parabola y2 = 6x. The locus of its circumcentre, is (25th Jan 2023 Shift 2)
(1) 4y2–18y+3x+18=0
(2) 4y2+18y+3x+18=0
(3) 4y2–18y–3x+18=0
(4) 4y2–18y–3x–18=0
8. If P(h, k) be a point on the parabola x = 4y2 , which is nearest to the point Q(0, 33), then the distance of P from the directrix of the parabola y2 = 4(x+y) is equal to (30th Jan 2023 Shift 1)
(1) 2 (2) 4
(3) 8 (4) 6
9. Let A be a point on the x -axis, common tangents are drawn from A to the curves x 2 + y 2 = 8 and y 2 = 16 x . If one of these tangents touches the two curves at Q and R, then (QR)2 is equal to (30th Jan 2023 Shift 2)
(1) 64 (2) 76 (3) 81 (4) 72
10. If the points of intersection of two distinct conics x 2 + y 2 = 4 b and, 22 2 1 16 xy b += lie on the curve y2 = 3x2, then 33 times the area of the rectangle formed by the intersection points is ______. (29th Jan 2024 Shift 1)
11. Let P( α , β )be a point on the parabola y2 = 4x. If P also lies on the chord of the parabola x 2 = 8 y whose mid point is 5 1, 4
. Then ( α –28)( β –8) is equal to ____.
(29th Jan 2024 Shift 2)
12. Let the tangent to the curve x2+2x–4y+9 = 0 at the point P(1,3) on it meet the y-axis at A Let the line passing through P and parallel to the line x–3y = 6 meet the parabola y2 = 4x at B. If B lies on the line 2x–3y = 8, then(A B)2 is equal to ____. (6th Apr 2023 Shift 1)
13. The ordinates of the points P and Q on the parabola with focus (3,0) and directrix x = –3 are in the ratio 3: 1. If R (α, β) is the point of intersection of the tangents to the parabola at P and Q then 2β α is equal to _____.
(8th Apr 2023 Shift 2)
14. Let a common tangent to the curves y2 = 4x and (x–4)2+y2 = 16 touch the curves at the points P and Q. Then (PQ)2 is equal to ____.
(10th Apr 2023 Shift 1)
15. Let the tangent to the parabola y2 = 12x at the point (3, α ) be perpendicular to the line 2x+2y = 3. Then the square of distance of the point (6, –4)from the normal to the hyperbola a 2x 2–9 y 2 = 9 α 2 at its point ( α -1, α +2) is equal to ______.
(11th Apr 2023 Shift 2)
16. Let an ellipse with centre (1,0) and latus rectum of length 1 2 have its major axis along x-axis. If its minor axis subtends an angle 60° at the foci then the square of the sum of the lengths of its minor axis and major axes is equal to _____.
(15th Apr 2023 Shift 1)
17. A triangle is formed by the tangents at the point (2,2) on the curves y2 = 2x and x2+y2 = 4x and the line x+y+2 = 0. If r is the radius of its circumcircle, then r2 is equal to ____.
(29th Jan 2023 Shift 2)
18. Let S be the set of all a∈N such that the area of the triangle formed by the tangent at the point P(b, c), b, c ∈ N, on the parabola y2 = 2ax and the lines x = b, y = 0 is 16 unit2, then aS a ∈ ∑ is equal to ___. (31st Jan 2023 Shift 2)
19. If the x -intercept of a focal chord of the parabola y2 = 8x+4y+4 is 3 , then the length of this chord is equal to ____.
(1st Feb 2023 Shift 2)
CHAPTER TEST – JEE MAIN
Section – A
1. The equation of parabola whose latus rectum is 2 units, axis is x+y-2 = 0 and tangent at the vertex is x–y+4 = 0 is given by
(1) (x+y-2)2 = 4 2 (x–y+4)2
(2) (x–y–4)2 = 4 2 (x+y-2)
(3) (x+y–2)2 = 2 2 (x–y+4)
(4) (x–y–4)2 = 2 2 (x–y+2)2
2. The focus of a parabola is (1,2) and the point of intersection of the directrix and axis is (2,3). Then the equation of the parabola is
(1) (x–1)2+(y–2)2 = 1 4 (x+y–5)2
(2) (x–1)2+(y–2)2 = 1 2 (x+y–5)2
JEE Advanced
20. Let E denote the parabola y2 = 8x. Let P (-2,4), and let Q and Q’ be two distinct points on E such that the lines PQ and PQ’ are tangents to E. Let F be the focus of E. Then which of the following statements is(are) TRUE?
(2021 P2)
(1) The triangle PFQ is a right-angled triangle
(2) The triangle QPQ ’ is a right-angled triangle
(3) The distance between P and F is 52 (4) F lies on the line joining Q and Q’ (Q. 21–22)
Consider the region R = {(x, y) ∈ : × x ≥ 0 and y2 ≤ 4–x}. Let F be the family of all circles that are contained in R and have centers on the x-axis. Let C be the circle that has largest radius among the circle in F . Let( α , β ), be a point where the circle Cmeets the curve y2 = 4–x. (2021 P1)
21. The radius of the circle C is _____.
22. The value of α is _____.
(3) (x–1)2+(y–2)2 = 1 5 (x+y–5)2
(4) (x–1)2+(y–2)2 = 1 25 (x+y–5)2
3. P is a point which moves in the xy plane such that the point P nearer to the centre of a square then any of the sides. The four vertices of the square are(±a, ±a). The region in which P will move is bounded by parts of parabolas of which one has the equation
(1) y2 = a2+2ax
(2) x2 = a2+2ay
(3) y2+2ax = a2
(4) all of these
4. The value of p such that the vertex of y = x2+2px+13 is 4 units above the x–axis is (1) 2 (2) ± 4 (3) 5 (4) ± 3
5. If two circles x2+y2–6x–6y+13 = 0 and x2+y2 –8y+9 = 0 interesects at A and B. The focus of the parabola whose directrix is line AB and vertex at (0,0) is
(1) 31 , 55 (2) 31 , 55
(3) 31 , 55 (4) 31 , 55
6. An equilateral triangle is inscribed in the parabola y2 = 4ax, where one vertex is at the vertex of the parabola. The length of side of triangle is (1) 83a (2) 43a (3) 33a (4) 23a
7. The point of contact of the line kx+y–4 =0 w.r.t the parabola y = x–x2 is (1) (–2, 2) (2) (2, –2) (3) (–2, 6) (4) (2, –6)
8. The point of the parabola y = x2+7x+2 which is closest to the line y = 3x-3 is (1) (2, 8)
(2) (2, –8)
(3) (–2, 8)
(4) (–2, –8)
9. Equation of the common tangent to the circle x2+y2 = 4ax and y2 = 4ax is
(1) x+y+a = 0
(2) x = 0
(3) x = a
(4) x–y+a = 0
10. If the line 33yx=− cuts the parabola y 2 = x +2 at P and Q , if A be the point ( 3 , 0) then |APAQ| is
11. Let PQ be a chord of the parabola y2 = 4x. A circle drawn with PQ as diameter passes through the vertex V of the parabola. If the area of triangle PVQ is 20 then coordinates of P are
(1) (–16, –8) (2) (–16, 8)
(3) (16, –8) (4) (8, 16)
12. A chord PP1 of a parabola cuts the axis of the parabola at O. The feet of the perpendiculars from P and P 1 on the axis are M and M 1 respectively. If V is the vertex then, VM , VO, VM1 are in
(1) AP (2) GP
(3) HP (4) AGP
13. If the segment intercepted by the parabola y2 = 4ax with the line lx+my+n = 0 subtends a right angle at the vertex then
(1) 4al+n = 0 (2) 4al+4am+n= 0
(3) 4am+n = 0 (4) al+n = 0
14. If a ≠ 0 and the line 2bx+3cy+4d = 0 passes through the points of intersection of the parabolas y2 =4ax and x2 = 4ay, then
(1) d2+(2b–3c)2 = 0
(2) d2+(3b+2c)2 = 0
(3) d2+(2b+3c)2 = 0
(4) d2+(3b–2c)2 = 0
15. If the tangents at t1, t2, t3 on y2 = 4ax make angles 30°, 45°, 60° with the axis then t1, t2, t3 are in
(1) AP (2) GP
(3) HP (4) AGP
16. If y1 and y2 are the ordinates of two points P and Q on a parabola and y3 is the ordinate of the point of intersection of the tangents at P and Q, then
(1) y1, y2, y3 are in AP
(2) y1, y3, y2 are in AP
(3) y1, y2, y3 are in GP
(4) y1, y3, y2 are in GP
17. Focal chord of y 2 = 16 x is a tangent to ( x –6) 2+ y 2 = 2, then possible values of the slopes are
(1) 1, –1
(2) –2, 2
(3) –2, 1 2
(4) 2 ,- 1 2
18. If the join of ends of the Latus rectum of x2 = 8y subtends an angle θ at the vertex of the parabola, then cos θ = (1) 4 5 (2) 3 2 (3) 5 3 (4) 5 1
19. The length of latus rectum of the parabola whose focus is 22 sin2,cos2 22 uu gg
and directrix is 2 2 u y g = is
(3)
(4)
Statement 2: The foot of perpendicular from the focus on any tangent of a parabola lies on the tangent at the vertex
(1) Statement- 1 is true, statement- 2 is true, statement- 2 is correct explanation for statement- 1
(2) Statement- 1 is true, statement- 2 is true, statement- 2 is not correct explanation for statement- 1
(3) Statement- 1 is true, statement- 2 is false.
(4) Statement- 1 is false, statement- 2 is true
Section – B
21. M is the foot of the perpendicular from a point P on the parabola y 2 = 8( x –3)to its directrix and S is the focus of the parabola, if SPM is an equilateral triangle, the length of each side of the triangle is _____.
22. The length of the double ordinate of the parabola y 2 –8 x +6 y +1 = 0 which is at a distance of 32 units from vertex is _____.
23. The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The road way which is horizontal and 100 m long is supported by vertical wires, attached to the cable, the longest wire being 30m and the shortest being 6m. The length of supporting wire attached to the road way 18 m from its middle is equal to _______.
24. The number of points on the curve y = || 1–ex|–2|from which two perpendicular tangents can be drawn to the parabola x 2 = –4 y is equal to ______.
25. If the tangents to the parabola y2 = 4ax make complementary angles with the axis of the parabola, then t1t2 = _____.
20. Statement 1: if a parabola touches the coordinate axes and has the focus (2,3), then the equation of the tangent at the vertex is 3x+2y = 5.
CHAPTER TEST – JEE ADVANCED
2019 P 1 Model
Section - A
[Multiple Option Correct MCQs]
1. Consider the parabola y2 = 4x. Let S be the focus of the parabola. A pair of tangents drawn to the parabola from the point P(-2, 1) meet the parabola at P1 and P2. Let Q1 and Q2 be points on the lines SP1 and SP2 respectively, such that PQ1 is perpendicular to SP1 and PQ2 is perpendicular to SP2 . Then which of the following is/are true?
2. If l,m are variable real numbers such that 5 l 2+6 m 2–4 lm +3 l =0, and the variable line lx + my =1 always touches a fixed parabola P(x, y) = 0 whose axis parallel to x-axis, then, which of the following is true?
(1) vertex of P(x, y) is 54 , 33
(2) latus rectum equal to 4
(3) directrix of P(x, y) is 3x+11 = 0
(4) if AB is a focal chord of P(x, y) = 0 then 111 2 SASB += , where S is the focus
3. If P 1 P 2 and Q 1 Q 2 two focal chords of a parabola are at a right angles, then
(1) Area of the quadrilateral P 1 Q 1 P 2 Q 2 is minimum when the chords are inclined at an angle 4 π to the axis of the parabola
(2) Minimum area of the quadrilaterl P1Q1P2Q2 is twice the area of the square on the latus rectum of the parabola
(3) Minimum area of quadrilateral P1Q1P2Q2 cannot be found
(4) Minimum area of quadrilateral P1Q1P2Q2 is thrice the area of the square on the latus rectum of the parabola
4. The tangent to the circle x2+y2 = 4 at a point P intersects the parabola y2 = 4x at points Q and R. Tangents to the parabola at Q and R intersect at S. If Q lies in the first quadrant such that PQ = 1 unit, then which of the following is/are TRUE? (where A x, A y denote x, y coordinates of A)
(1) Qx +Qy = 3 (2) 14 5 xy PP+=
(3) 5 4 xy RR+= (4) Q = (1, 2)
5. Equation of common tangent of y = x 2 , y = –x2+4x–4 is/are
(1) y = 4(x–1)
(2) y = 0
(3) y = –4(x–1) (4) y = –30x–50
6. The curves x 2+ y 2+6 x –24 y +72 = 0 and x 2 –y2+6x+16y–46 = 0 intersect in four point P, Q, R, and S lying on parabola. Let A be the focus of the parabola, then (1) Equation of directrix is y+1 = 0 (2) length of Latus rectum is 8 (3) vertex of the parabola is at (–3, 1) (4) coordinates of A are (–3, 2)
Section - B
[Numerical Value Questions]
7. Let L1: x+y = 0 and L2 : x–y = 0 are tangent to a parabola whose focus is S (1,2). If the length of latus rectum of the parabola can be expressed as m n (where m and n are coprime) then the value of (m+n) is _____.
8. If the locus of centres of a family of circles passing through the vertex of the parabola y 2 = 4 ax and cutting the parabola orthogonally at the other point of intersection is 2 y 2(2 y 2+ x 2–12 ax ) = ax ( kx –4a)2, then the value of k is ____.
9. If the normals at the points P, Q, R on the parabola y2 = 4x meet in the point (h, k). If the centroid and orthocenter of the triangle PQR is (x1, y1) and (x2, y2) then find the value of 3x1–2x2 is _____.
10. A series of chords is drawn to the parabola y 2 = 4 ax , so that their projections on a straight line which is inclined at an angle α to the axis are all of constant length c if the locus of their middle point is the curve. (y2 –ax)(ycosα+2 asin α)2+a2c2 = 0. Then λ _____.
11. If the locus of intersections of tangents to the parabola y2 = 4ax which intercept a fixed length I on the directrix is (y2 –λax)(x+a)2 = l2x2. Then λis ______.
12. The radius of circle which passes through the focus of parabola x2 = 4y and touches it at point (6,9) is 10 k , then k = _____.
13. An equilateral triangle ABC is inscribed in the parabola y = x 2 and one of the side of the equilateral triangle has the gradient 2. If the sum of x-coordinates of the vertices of the triangle is a rational in the form p q where p and q are coprime, then the value of () 7 pq + is ______.
14. Two circles with centres at vertex and focus of parabola y2–2y–8x = 15 have one direct common tangent is px+4y+q = 0 then p–q equals _____.
Section - C [Single Option Correct MCQs]
15. ABCD and EFGC are squares and the curve y = kx passes through the origin, D and the points B and F. The ratio FG BC is
16. The length of latus-rectum of the parabola whose parametric equation is x = t2+t+1 and y = t2-t+1, where t ∈ R-{0} is equal to
(1) 2 (2) 4
(3) 8 (4) 10
17. A point P on a parabola y2 = 4x, the foot of the perpendicular from it upon its directrix, and the focus of the parabola are the vertices of an equilateral triangle. The area of the equilateral triangle is “k“ sq. units, then the value of [k] = Where [.] denotes greatest integer function
(1) 5 (2) 8 (3) 6 (4) 4
18. Let 3x – y – 8 = 0 be the equation of tangent to a parabola at the point (7, 13). If the focus of the parabola is at (-1,-1) then its directrix is
Learning about ellipse equations helps us understand how ellipses work. This knowledge is important in many areas like space exploration, building things, and figuring out how light moves. It help us solve problems and come up with new ideas in science and engineering.
15.1 INTRODUCTION TO ELLIPSE
The locus of a point whose distances from a fixed point and a fixed straight line are in constant ratio e, where 01 e << , is called an ellipse.
Here, the fixed point is called focus, fixed line is called directrix, and e is called eccentricity. A second degree non-homogeneous equation 222220 axhxybygxfyc +++++= represents an ellipse if 2 0 hab−< and 222 20abcfghafbgch+−−−≠ .
15.1.1 Equation of Ellipse in the Standard Form
The equation of an ellipse in the standard form is 2 2 22 1; xyab ab +=> .
If () , xy is a point on the curve, then ()() ,,, xyxy and () , xy also lie on the curve. So, the curve is symmetric about both coordinate axes and symmetric about origin also. If 0 y = , then xa =± . Hence, the curve cuts the x -axis at two points () ,0Aa and () ,0Aa ′ .
The line segment through the foci of the ellipse with its end points on the ellipse is called major axis. The line segment AA′ is called major axis and its length is 2 a.
If 0 x = , then yb =± . Therefore, the curve cuts the y -axis at two points () 0, Bb and () 0, Bb ′ . The line segment through the centre of the ellipse and perpendicular to the major axis with its end points on the ellipse is called minor axis. The line segment BB ′ is called minor axis and its length is 2b .
From the equation of ellipse, 22 b yax a =±− , it means that y is real for all axa−≤≤ . Similarly, x is real for all byb−≤≤ Therefore, the ellipse is a bounded curve, and it lies within the rectangle formed by the lines , xayb =±=± .
The focus of the ellipse is () ,0 Sae and the corresponding directrix is a x e = . By the symmetry of the curve, there exists second
focus () ,0 Sae ′ and corresponding directrix
a x e =− . Here, 22ab e a =
For the ellipse, there exists two foci, two directrices, two latus recta, one major axis, one minor axis, two vertices, one centre, etc. A chord passing through either of the foci is called a focal chord.
Important points:
1. The locus of a point, the sum of whose distances from two fixed points is a constant k , provided the distance between the fixed points is less than k , is an ellipse.
2. Let P be any point on the ellipse. Then, 2 SPSPa ′ += where 2a is the length of the major axis and , SS ′ are foci.
3. The angle made by the chord joining the end points of major axis and minor axis of ellipse with x-axis is 1 22 sin b ab
4. A chord passing through any point P on the ellipse and perpendicular to major axis is called double ordinate of the point P
5. A focal chord of an ellipse perpendicular to the major axis of the ellipse is called latus rectum. There will be two latus recta in an ellipse.
6. If PSQ is a focal chord of the ellipse, then 112 SPSQSL += , where SL is semi latus rectum.
7. Two ellipses are said to be similar if they have same eccentricity.
1. Find the equation of the ellipse whose focus is ()0,3, eccentricity is 3 5 and directrix is 3250 y −=
Sol: Directrix is 3250 y −= , parallel to x-axis and focus lies on y-axis.
The required ellipse is a vertical ellipse; focus is ()() 0,0,3 be = .
Hence, 3 35 5 bb =⇒= and 22 2 325 4 55 baa ea b =⇒=⇒=
Therefore, the equation of the ellipse is 2 2 1 1625 xy+=
Try yourself:
1. If the equation 2 2 1 14 xy aa += represents an ellipse, then find the range of a Ans: 1 a <
TEST YOURSELF
1. The equation of the ellipse having focus (–1, 1), 1 2 e = , and directrix x – y + 3 = 0 is (1) 5x2 + 2xy + 5y2 + 10x – 10y + 5 = 0 (2) 7x2 + 2xy + 7y2 + 10x – 10y + 7 = 0 (3) 3x2 + 2xy + 3y2 + 5x – 5y + 5 = 0 (4) 9x2 + 2xy + 9y2 + 15x – 15y + 10 = 0
2. If PSP' is a focal chord of the ellipse 2 2 1 79 xy+= , then ' ' . SPSP SPSP = + (1) 7 3 (2) 7 6 (3) 9 7 (4) 7 3
3. If the minor axis of an ellipse subtends an angle of 60° at each focus, then e = (1) 3 2 (2) 1 2 (3) 2 3 (4) 1 3
4. If the equation 2 2 y x 1 10kk4 += represents an ellipse, then (1) k < 4 (2) k > 10 (3) 4 < k < 10 (4) ()() k4,77,10 ∈∪
5. If a conic has latus rectum of length 1, focus at (2, 3), and the corresponding directrix is x + y – 3 = 0, then the conic is
(1) a parabola
(2) an ellipse
(3) a hyperbola
(4) a rectangular hyperbola
6. If A = (1, 2), B = (3, –2), and P moves in the plane such that AP+BP = 7, then the locus of P has two axes of symmetry. Their equations are
(1) x–2y+3 = 0, 2x+y = 4
(2) 2x+y = 4, x–2y = 2
(3) x–2y = 2, x–y+1 = 0
(4) x–2y = 7, 2x+y = 4
7. If the distance between the foci of an ellipse is equal to half of its minor axis, then eccentricity is (1) 1 2 (2) 1 3 (3) 1 2 (4) 1 5
8. If the distance between the directrices is 8 times the distance between the foci, then e = (1) 1 2 (2) 1 22 (3) 1 4 (4) 1 8
Answer Key
(1)
15.2 DIFFERENT FORMS OF ELLIPSE
Ellipses with major axes aligned along coordinate axes or parallel to them exhibit unique properties. When the major axis aligns with the x-axis, it’s called a horizontal; when it aligns with the y-axis, it’s a vertical.
15.2.1 Equation of Ellipse Whose Major Axis Is along Coordinate Axes
1. Equation of ellipse in standard form is 2 2 22 1, xy ab+=() 0 ab>>
This is an ellipse having major axis along the x -axis and centre at origin. This is called horizontal ellipse.
2. For the ellipse 2 2 22 1 xy ab += , () 0 ab>> ,
i. Centre is () 0,0 C
ii. The major axis is along x -axis and equation of major axis is y = 0.
iii. The minor axis is along y -axis and equation of minor axis is x = 0
iv. The length of major axis is 2a and the length of the minor axis is 2b
v. The ends of the major axis are vertices ()() ,0,,0AaAa ′ .
vi. The ends of the minor axis are ()() 0,,0, BbBb ′
vii. Eccentricity is 22ab e a =
viii. Foci are ()() ,0,,0SaeSae ′ and the distance between the foci is 2ae
ix. If P is any point on the ellipse, then 2 SPSPa ′ +=
x. The length of the latus rectum is 2 2b a and the equations of the latus recta are , xaexae ==−
xi. Ends of the latus rectum are 2 , b ae a
xii. Equations of directrices are a x e =± and distance between the directrices is 2a e
3. Equation of ellipse 2 2 22 1 xy ab += , () 0 ab<<
This is an ellipse having major axis along the y -axis and centre at origin. This is called vertical ellipse.
4. For the ellipse 2 2 22 1 xy ab += , () 0 ab<<
i. Centre is () 0,0 C
ii. The major axis is along y -axis and equation of major axis is x = 0.
iii. The minor axis is along x -axis and equation of minor axis is y = 0.
iv. The length of major axis is 2b and the length of the minor axis is 2a
v. The ends of the major axis are vertices ()() 0,,0, BbBb ′
vi. The ends of the minor axis are ()() ,0,,0AaAa ′
vii. Eccentricity is 22ba e b =
viii. Foci are ()() 0,,0, SbeSbe ′ and the distance between the foci is 2 SSbe ′ =
ix. If P is any point on the ellipse, then 2 SPSPb ′ +=
x. The length of the latusrectum is 2 2a b and the equations of the latusrecta are , ybeybe ==−
xi. Ends of the latus recta are 2 , abe b
±±
xii. Equations of directrices are b y e =±
2. Find the equation of the ellipse whose vertices are ()() 5,0,5,0 and one of the foci lies on the line 359 xy−= .
Sol: The vertices ()() 5,0,5,0 lie on the x-axis, so that the ellipse is in standard form, and the equation of the ellipse can be taken as 2 2 22 1 xy ab += .
Here, 5 a = . The foci () ,0 ae ± lies on the line 359 xy−=
So, () 3 350953 5 aeee −=⇒=⇒=
Hence, 222 2594 baae=−=−=
Therefore, the equation of the required ellipse is 2 2 1 2516 xy+=
Try yourself:
2. Find the equation of the ellipse in the standard form which is passing through the point () 2,1 and having eccentricity 1 2 e =
Ans: 22 3416 xy+=
15.2.2 Equation of Ellipse Having Axes Parallel to Coordinate Axes
1. ()() 22 22 1 xhyk ab += , () 0 ab>>
This is an ellipse having major axis parallel to the x-axis and centre at () , hk . This is called horizontal ellipse.
Minor axis
Majoryaxis = k x = h
2. For the ellipse ()() 22 22 1 xhyk ab += , () 0 ab>>
i. Centre is () , Chk
ii. The major axis is parallel to x-axis and equation of major axis is 0 yk−= .
iii. The minor axis is parallel to y-axis and equation of minor axis is 0 xh−= .
iv. The length of major axis is 2a and the length of the minor axis is 2 b
v. The ends of the major axis are vertices ()() ,,, AahkAahk ′ +−+
vi. The ends of the minor axis are ()() ,,, BhbkBhbk ′ +−+
vii. Eccentricity is 22ab e a =
viii. Foci are ()() ,,, ShaekShaek ′ +− .
ix. The length of the latus rectum is 2 2b a and the equations of the latus rectum are , xhaexhae −=−=−
x. Equations of directrices are xha e −=±
3. ()() 22 22 1 xhyk ab += , () 0 ab<<
This is an ellipse having major axis parallel to the y-axis and centre at () , hk . This is called vertical ellipse. y
Major axis y = k x = h x B C A Z S S’ Z’ A’ B’ L’ L
Minor axis
4. For the ellipse ()() 22 22 1 xhyk ab += ,
() 0 ab<<
i. Centre is () , Chk
ii. The major axis is parallel to y-axis and equation of major axis is 0 xh−= .
iii. The minor axis is parallel to x-axis and equation of minor axis is 0 yk−= .
iv. The length of major axis is 2b and the length of the minor axis is 2 a.
v. The ends of the major axis are vertices ()() ,,, BhbkBhbk ′ +−+
vi. The ends of the minor axis are ()() ,,, AahkAahk ′ +−+
vii. Eccentricity is 22ba e b =
viii. Foci are ()() ,,, ShkbeShkbe ′ +−
ix. The length of the latus rectum is 2 2a b and the equations of the latus rectum are −=−=− , ykbeykbe .
x. Equations of directrices are −=± ykb e
3. Find the equation of the ellipse whose vertices are () 4,1 and () 6,1 one of the focal chords is 220xy−−=
Sol: The vertices () 4,1 and () 6,1 lie on the line 10 y −= , and the distance between the vertices is 2105 aa=⇒=
Centre of the ellipse is () 1,1 , which is midpoint of vertices.
Since the ellipse is horizontal ellipse, the foci are () 1,1 ae ±
The point () 1,1 ae + lies on 220xy−−= ,
so that 3 12203 5 aeaee +−−=⇒=⇒=
Hence, ()222 16 12516 25 bae =−==
Therefore, the equation of the ellipse is ()() 22 11 1 2516 xy +=
Try yourself:
3. Find the equation of the ellipse having vertices ()() 2,2,2,4 and 1 3 e = .
Ans: )()( 22 21 1 89 xy +=
15.2.3 Equation of an Ellipse Referred to Two Perpendicular Lines
Let the minor and major axes of the ellipse be along two mutually perpendicular lines
1111 0 Laxbyc≡++= and 2112 0 Lbxayc≡−+= , respectively. Let , ab be the lengths of semimajor axis and semi-minor axis, respectively. () ab >
The equation of the ellipse is 22
111112
22 22 11 11 22 1 ++−+ ++ += axbycbxayc abab ab
The characteristics of the above ellipse are as follows:
1. The centre of the ellipse is the point of intersection of two lines 1 0 L = and 2 0 L =
2. If ab > , then
i. the major axis is along the line 2 0 L = and the minor axis is along the line 1 0 L =
ii. foci are the points of intersection of 22 111 0 Laeab±+= and 2 0 L =
iii. equations of directrices are 22 1111 0 a axbyab e +±+=
3. If ab < , then
i. the major axis is along the line 1 0 L = and the minor axis is along the line 2 0 L =
ii. foci are the points of intersection of 1 0 L = and 22 211 0 Lbeab±+=
iii. equations of directrices are 22 1111 0 b bxayab e −±+=
4. Find the major axis, minor axis, centre, and eccentricity of the ellipse
()() 22 421922180 xyxy −++++=
Sol: Equation of the ellipse is ()() 22 2122 1 4520 xyxy −+++ +=
The major axis is 220 xy++= and the minor axis is 210xy−+= .
Centre is the point of intersection of the above two lines, so it is () 1,0 and the eccentricity is === 22 945 93 ab e a
Try yourself:
4. Find the equation of the ellipse whose axes are of lengths 6 and 26 . Their equations are 330xy−+= and 310 xy+−= , respectively Ans: )()( +−−+ += 22 3133 1 96 xyxy
TEST YOURSELF
1. A point P moves such that the distance from the point (2, 0) is always 1/3 of its distance from the line x – 18 = 0. If the locus of point P is conic, the length of its latus rectum is (1) 16 3 (2) 32 3 (3) 8 3 (4) 15 4
2. If focus is (4, 0), 1 e, 2 = directrix is x–16 = 0, then equation of the ellipse is (1) 2 2 1 169 xy+= (2)
5. If foci are (0, ± 3), e = 3 4 , equation of the ellipse is (1) 2 2 1 716 xy+= (2) 2 2 1 916 xy+= (3) 2 2 1 918 xy+= (4) 2 2 1 1625 xy+=
6. If centre is (1, 2), axes are parallel to coordinate axes, distance between the foci is 8, 1 e 2 = , then equation of the ellipse is
3. For an ellipse with eccentricity 1/2 and centre at origin, if one directrix is x = 4, then the equation of ellipse is (1) 3x2+4y2 = 1 (2) 3x2+4y2 = 12 (3) 4x2+3y2 = 1 (4) 4x2+3y2 = 12
4. Axes are coordinate axes, A and L are the ends of major axis and latus rectum. Area of 1 8 ., e 2 OALsqunits ∆== . Then, equation of the ellipse is (1)
2 1 168 xy+= (2)
2 1 3216 xy+= (3)
2 1 6432 xy+= (4)
2 1 84 xy+=
7. The eccentricity of the ellipse x2+4y2+2x+16y+13 = 0 is (1) 3
8. If the eccentricity of the ellipse is 2 2 22 1 1 12 6 xybe aa += ++ , then length of the latus rectum of the ellipse is (1) 5 6 (2) 10 6 (3) 8 6 (4) 5 2
9. The equation of one of the latus recta of ()() 22 x3y5 1 59 += is
(1) y–3 = 0 (2) x–1 = 0 (3) y = 6 (4) x = 5
10. In a model, it is shown that an arch of a bridge is semi-elliptical with major axis horizontal. If the length of the base is 9 m and the highest part of the bridge is 3 m from the horizontal, the best approximation of the height of the arch, 2 m from the centre of the base, is
(1) 11 m 4 (2) 8 m 3 (3) 7 m 2 (4) 2 m
11. The equation of the normal to the ellipse x2+3 y2 = 144 at the positive end of the latus rectum is
(1) 3282 xy−=
(2) 3282 xy+=
(3) 3222 xy+=
(4) 3222 xy−=
Answer Key
(1) 2 (2) 3 (3) 2 (4) 2
(5) 1 (6) 1 (7) 1 (8) 2 (9) 1 (10) 2 (11) 1
15.3 TANGENT AND NORMAL
Chords on an ellipse are line segments connecting any two points on the curve. When extended as lines, as one point approaches the other along the curve and the chord becomes a tangent to that point on the curve. Normals are perpendicular lines to tangents at their point of contact.
The equation of the chord joining two points () 11 , Pxy and () 22 , Qxy on the ellipse
0 S = is 1212SSS += .
15.3.1 Tangent Line to the Ellipse
Position of a line with respect to the ellipse:
Consider the ellipse 2 2 22 1 xy ab += and the line ymxc =+ . The line will
i. intersects the ellipse, if 2222 camb <+
ii. touches the ellipse, if 2222 camb =+
iii. does not intersect the ellipse, if 2222 camb >+
The equation of tangent to the ellipse 0 S = at () 11 , Pxy is 1 0 S = .
The slope form of tangent to the ellipse 2 2 22 1 xy ab += is 222 ymxamb =±+ and its point of contact is 22 , amb cc
, where 222 camb =±+
The condition for the line 0 lxmyn++= t o be tangent to the ellipse 2 2 22 1 xy ab += is 22222 albmn += and the point of contact is 22 , albm nn
Important points:
The feet of the perpendicular drawn from either of the foci to any tangent to the ellipse lie on a circle concentric with the ellipse. This circle is called the auxiliary circle of the ellipse.
1. If ab > , then the equation of auxiliary circle of the ellipse 2 2 22 1 xy ab += is 222 xya += . If ab < , then the equation of the auxiliary circle is 222 xyb += .
2. Product of the perpendiculars from foci on any tangent to the ellipse 2 2 22 1 xy ab += is 2 b . (a > b)
3. If tangent at a point P to the ellipse intersects the major axis at T . If N is foot of the perpendicular from P is on major axis, then the circle having NT as diameter intersects the auxiliary circle orthogonally.
4. The tangents at the extremities of the latus rectum of an ellipse intersect on corresponding directrix.
5. The tangents at the extremities of the focal chord of an ellipse intersect on corresponding directrix.
6. The product of the perpendiculars from the foci of any tangent to an ellipse is equal to the square of the semi-minor axis and the feet of these perpendiculars lie on the auxiliary circle
7. If any tangent to an ellipse intersects the tangents at the vertices at T and T ′ , then the circle considering TT ′ as diameter passes through the foci.
8. The portion of the tangent to an ellipse between the point of contact and the directrix subtends right angle at corresponding focus.
9. The circle on any focal distance of ellipse as diameter touches the auxiliary circle.
10. Lines joining centre to the foot of perpendicular from a focus on any tangent at point P is parallel to the line joining other focus to the point of contact P
5. Find the equation of tangent to the ellipse
22 236 xy+= which makes an angle 30° with major axis.
Sol: Equation of the ellipse is 2 2 1 32 xy+=
The slope of the tangent is 1 tan30 3
of tangent in the slope form
Try yourself:
5. If 34122 xy+= is a tangent to the ellipse 2 2 2 1 9 xy a += for some aR ∈ , then find the distance between the foci of the ellipse.
Ans: 27
15.3.2 Normal Line to the Ellipse
The equation of normal to the ellipse
11 , Pxy is
Equation of normal drawn at 2 , b ae a
is 3 xeyae −=
The condition for the line 0 lxmyn++=
to be a normal to the ellipse 2 2 22 1 xy ab += is ()2 22 22 222 abab lmn += .
The slope form of normal to the ellipse 2 2 22 1 xy ab += is () 22 222 mab ymx abm =± + .
If the normal at P meets major axis at G, then SG=e(SP), SG/S'G = SP/S'P.
Let S and S ′ be two foci of an ellipse. The normal at P to the ellipse is internal angular bisector of SPS ∠′ and tangent at P to the ellipse is external angular bisector of SPS ∠′ .
Reflection property:
If an incoming ray passes through one focus (S) and strikes the concave side of ellipse, then the reflected ray passes through the other focus.
Therefore, the equation of the tangent is
xy−±=
6. If the normal at one end of a latus rectum of the ellipse 2 2 2 1 32 xy b += passes through one end of the minor axis, then find the value of 4 2 1 e e
Sol: Equation of the normal is 2 2 22 2 axbyab aeb a −=−
Given: It is passing through the point () 0, b
Hence, 22abab =− . Substitute () 2 1 bae =− and then simplify.
It implies that 4 2 1 1 e e = Try yourself:
6. Find a point on the ellipse 22337xy+= when the normal is parallel to the line 652 xy−=
Ans: (5, 2)
TEST YOURSELF
1. The values that m can take so that the straight line y = 4 x + m touches the curve x2+4y2 = 4 is
(1) 45 ± (2) 60 ±
(3) 65 ± (4) 72 ±
2. The equation of tangent to the ellipse 2x2+3y2 = 6, which makes an angle 30° with the major axis, is
(1) 330xy−±= (2) 330xy+±=
(3) 330yx−±= (4) y3x30 +±=
3. The locus of the midpoint of the portion of a tangent to the ellipse 2 2 22 1 xy ab += included between the coordinate axes is
(1) 22 22 ab 1 xy += (2) 22 22 2 ab xy += (3) 2 2 22 4 xy ab += (4) 22 22 4 ab xy +=
4. The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latus recta to the ellipse 2 2 1, 95 xy+= is
(1) 27 4 (2) 18 (3) 27 2 (4) 27
5. The equation of the normal to the ellipse x2+3y2 = 144 at the positive end of the latus rectum is
(1) 3282 xy−=
(2) 3282 xy+= (3) 3222 xy+=
(4) 3222 xy−=
6. The equation of common tangent to the circle x 2 + y 2 = 16 and to the ellipse 2 2 1 494 xy+= is
(1) 45 yx=+
(2) 53 yx=+
(3) 1124 yx=+
(4) 112415 yx=+
7. The equation of the locus of the foot of the perpendicular drawn from the centre of the ellipse 2 2 22 1 xy ab += to any tangent of the ellipse is
(1) (x2+y2)2 = a2x2+b2y2
(2) (x2 –y2)2 = a2x2+b2y2
(3) (x2+y2)2 = a2x2 –b2y2
(4) (x2 –y2)2 = a2x2 –b2y2
8. C is the centre of the ellipse 2 2 22 1 xy ab += and L is an end of a la tus rectum. If the normal at L meets the major axis at G, then CG =
(1) ae (2) 2ae2 (3) ae3 (4) a2e2
9. The radius of the largest circle with centre (1, 0) that can be inscribed in the ellipse x2+4y2 = 16 is
(1) 11 r 3 = (2) r 3 22 =
(3) r 22 5 = (4) r 21 4 =
10. If F1 and F2 are the feet of the perpendiculars from the foci S 1 and S 2 of the ellipse 2 2 1 2516 xy+= on the tangent at any point P on the ellipse, then S1F1+S2F2 (1) =5 (2) =16 (3) ≥ 8 (4) < 8
Answer Key
(1) 3 (2) 1 (3) 4 (4) 4
(5) 1 (6) 4 (7) 1 (8) 3
(9) 1 (10) 3
15.4 PARAMETRIC EQUATIONS
Let P be a point on the ellipse with centre at C and NP be the ordinate of the point P . Let NP meet the auxiliary circle in P ′ . If measure of angle NCP ′ is θ , then θ is called the eccentric angle of P . The point P ′ is called corresponding point of P
The area of the ellipse 2 2 22 1 xy ab += is p ab. The maximum area of the rectangle that can be inscribed in the above ellipse is 2ab and the sides are 2,2ab
The equation of the chord joining two points , αβ on the ellipse 2 2 22 1 xy ab += is cos sincos 222 xy ab αβαβαβ ++− +=
.
The condition for the chord joining , αβ to be a focal chord is
i. αβαβ −+ =±
cos cos 22 e
ii. αβ−+
tantan(or)11 2211 ee ee
iii. () sinsin sin e αβ αβ + = +
If the chord joining the points α and β on the ellipse S = 0 cuts the major axis at a distance ‘ d ’ units from the centre, then tantan 22 da da αβ
Let θ be the eccentric angle of a point () , Pxy on the ellipse 2 2 22 1 xy ab += . Then, cos,sinxaybθθ== , where 02θπ≤< .
The parametric equations of the ellipse 2 2 22 1 xy ab += are cos,sinxaybθθ== .
The point () cos,sinPabθθ is a point on the ellipse 2 2 22 1 xy ab += , whose eccentric angle is θ . denoted by P( θ )
The corresponding point of () cos,sinPabθθ is () cos,sinPaa′θθ
Eccentric angles of the extremities of latus rectum of an ellipse 2 2 22 1 xy ab += are 1 tan b ae
.
The equation of tangent at () P θ to the ellipse 2 2 22 1 xy ab += is cossin1xy ab θθ+= .
Point of intersection of tangents at (a cos a, b sin a)( a cos b, b sin b) is cossin 22 , coscos 22 ab
The equation of the normal to the ellipse
2 2 22 1 xy ab += at () P θ is 22 cossin axbyab θθ −=− .
The ratio of area of triangle PQR inscribed in an ellipse to the area of the triangle formed by the corresponding points of ,, PQR is constant b/a
The focal distance of a point ()() 11,cos,sinPxyab=θθ is
1 cos SPaexaae=−=−θ and
1 cos SPaexaae ′θ =+=+
15.4.1 Concyclic Points, Co-normal Points on Ellipse
If a circle cuts an ellipse in four real or imaginary points, then the sum of the eccentric angles of these four concyclic points on the ellipse is an even multiple of p .
The points on the ellipse, at which the normals drawn to the ellipse passes through a given point, are called co-normal points.
Atmost four normals can be drawn from any point to the ellipse and the sum of the eccentric angles of their feet is an odd multiple of p
The sum of the eccentric angles of the feet of the normals to an ellipse through a point is an odd multiple of p . If 1234 ,,, θθθθ are feet of normals to the ellipse, then () 1234 21 n θθθθπ +++=+ .
If 1234 ,,, θθθθ are eccentric angles of four co-normal points on the ellipse
2 2 22 1 xy ab += , then () 12 cos0 θθ ∑+= and () 12 sin0 θθ ∑+=
7. If , αβ are the eccentric angles of the extremities of a focal chord of the ellipse
2 2 1 4 x y += , then what is the value of 3cos 2 αβ +
?
Sol: We know that coscos 22 e αβαβ −+ =
and 22 3 2 ab e a == Hence, 3cos 2 αβ + is 2cos 2 αβ
Try yourself:
7. If the equation of the normal to the ellipse 2 2 1 169 xy+= at the point whose eccentric angle 6 θπ = is 8633 xyk −= then find the value of k
Ans: 7
TEST YOURSELF
1. The point 4 P π
lies on the ellipse
2 2 1 42 xy+= , whose foci are S and S'. The equation of external angular bis ector of ∠ SPS' of triangle SPS' is
(1) 222xy+=
(2) 23312 xy+=
(3) 341220 xy++=
(4) 1220 xy++=
2. If S and S ' are the foci of the ellipse 2 2 1 2516 xy+= and P is any point on it , f(θ) = SP.S'P, then the range of f(θ) is
(1) 9 ≤ f( θ ) ≤ 16
(2) 9 ≤ f( θ ) ≤ 25
(3) 16 ≤ f( θ ) ≤ 25
(4) 1 ≤ f( θ ) ≤ 16
3. If the chord joining two points, whose eccentric angles are α and b, cuts the major axis of an ellipse 2 2 22 1 xy ab += at a distance c from the centre, then tantan 22 αβ =
(1) ca ca + (2) ca ca + (3) c a (4) c + a
4. If the normal at the point P(θ) to the ellipse 5x2 + 14y2 = 70 intersect it again at the point Q(2 θ ), then cos θ = (1) 2 3 (2) 2 3 (3) 1 3 (4) 1 3
5. The line 12x cosθ + 5y sinθ = 60 is tangent to which of the following curves? (1) x2 + y2 =169
(2) 144x2 + 25y2 = 3600
(3) 25x2 +12y2 = 3600 (4) x2 + y2 = 60
6. If P( θ ) and 2 Q πθ + are two points on the ellipse 2 2 22 1, xy ab += then locus of the mid – point of PQ is (1) 2 2 22 1 2 xy ab += (2) 2 2 22 4 xy ab += (3) 2 2 22 2 xy ab += (4) None of these
7. The area of locus of point of intersection of tangents of 2 2 22 1 xy ab += at two points whose eccentric angles differ by 2 π is (1) πab (2) 2πab (3) 4πab (4) 8πab
9. The equation of the normal to the ellipse 2 2 1 42 xy+= at the point whose eccentric angle is 4 π is (1) 222xy+= (2) 21 xy−= (3) 20xy−= (4) 23 xy+=
10. If the extremities of a focal chord are 5 12 π and 12 −π , then e = (1) 1 2 (2) 2 3 (3) 3 4 (4) 3 2
11. If P(θ) is a point on the ellipse 2 2 22 1 xy ab += (a > b), then its corresponding point is (1) (a cos θ, b sin θ) (2) (a cos θ, –b sin θ) (3) (a cos θ, a sin θ) (4) (–a cos θ, a sin θ)
12. Three distinct points A, B, and C are given in the 2-dimensional coordinate plane such that the ratio of the distance of any one of them from the point (1, 0) to the distance from the point (–1, 0) is equal to 1 3 Then, the circumcentre of the triangle ABC is at the point (1) (0, 0) (2) 5 ,0 4
(3) 5 2 ,0
8. A tangent is drawn at the point () 33cos,sin0 2 θθθπ << of an ellipse 2 2 1 271 xy+= . The least value of the sum of the intercepts on the coordinate axes by this tangent is attained at θ = (1) 6 π (2) 3 π (3) 8 π (4) 4 π
(4) 5 3 ,0
Key
15.5 PAIR OF TANGENTS
The combined equation of pair of tangents drawn from a point () 11 , Pxy to the ellipse 0 S = is 2 111SSS =⋅
If 12 , mm are the slopes of the two tangents drawn from a point () 11 , Pxy to the ellipse, then 12 , mm are the roots of the equation
()() 222 22 1 111 20xamxymyb−−+−= .
If 12 , mm are the slopes of the two tangents drawn from a point () 11 , Pxy to the ellipse, then 11 12 22 1 2 xy mm xa += and 22 1 12 22 1 yb mm xa =
If θ is angle between two tangents, then 11 2222 11 2 tan abS xyab θ = +−− .
The locus of the point of intersection of two tangents to the ellipse
2 2 22 1 xy ab += , which makes an angle a with one another, is ()() 2 22222222222 4 cot xyabbxayab +−−=+−α .
Point of intersection of the tangents at two points on the ellipse
2 2 22 1 xy ab += , whose eccentric angles differ by a right angle, lies on the ellipse 2 2 22 2 xy ab += .
Director Circle:
The locus of the point of intersection of perpendicular tangents to an ellipse is a circle. This circle is called director circle . The equation of director circle to the ellipse 2 2 22 1 xy ab += is 2222 xyab +=+ .
The equation of director circle to the ellipse ()() 22 22 1 xhyk ab += is ()() 22 22xhykab −+−=+ .
Chord of Contact:
The line joining the points of contact of the tangents to an ellipse 0 S = , drawn from an external point P , is called the chord of contact of P with respect to the ellipse 0 S = .
The equation of chord of contact of () 11 , Pxy with respect to the ellipse 0 S = is 1 0 S = .
Midpoint of Chord:
The equation of a chord of ellipse 0 S = whose midpoint () 11 , Pxy is 111SS =
The midpoint of the chord 0 lxmyn++= of the ellipse 2 2 22 1 xy ab += is 22
The locus of midpoint of normal chords of an ellipse 2 2 22 1 xy ab += is
The locus of midpoints of focal chords of the ellipse 2 2 22 1 xy ab += is 2 2 22 y xex aba += .
Diameter of an Ellipse:
The locus of midpoint of a system of parallel chords of an ellipse is called diameter of the ellipse. Diameter always passes through the centre of the ellipse.
Points to remember:
1. Equation of diameter corresponding to a system of parallel chords with slope m of an ellipse 2 2 22 1 xy ab += is 2 2 b yx am =
2. The tangents at the extremities of any diameter is parallel to the chords bisected by that diameter.
3. Two diameters of an ellipse are said to be conjugate diameters if each bisects the chords parallel to the other.
4. If the straight lines 1 ymx = and 2 ymx = are conjugate diameters of an ellipse
2 2 22 1 xy ab += , then 2 12 2 b mm a =− .
5. The tangents at the extremities of a pair of conjugate diameters form a parallelogram with constant area equal to 4ab .
6. The eccentric angles at the ends of a pair of conjugate diameters of an ellipse differ by a right angle.
8. Find the slopes of the tangents drawn from () 4,1 to the ellipse 2226.xy+=
Sol: If 12 , mm are the slopes of tangents drawn from () 11 , xy to the ellipse 2 2 22 1 xy ab += , then () () 11 12 22 1 241 2 4 5 46 xy mm xa +=== and 22 2 1 12 22 1 131 5 46 yb mm xa ===−
Therefore, the slopes are 1 1, 5
Try yourself:
8. If the chords of contact of () 11 , Pxy and () 22 , Qxy , with respect to the ellipse
2 2 22 1 xy ab += , are at right angles, then find the value of 12 12 xx yy Ans: 4 4 a b
TEST YOURSELF
1. The number of tangents to
94 xy+= through (3, 2) is (1) 0
(2) 1
(3) 2 (4) 3
2. The point of intersection of the perpendicular tangents drawn to the ellipse 4x2+9y2 = 36 lies on the circle
(1) x2+y2 = 13
(2) x2 –y2 = 5
(3) x+y = 5
(4) 2 2 1 94 xy+=
3. Angle between the tangents drawn from the point (5, 4) to the ellipse 2 2 1 2516 xy+= is
(1) 45°
(2) 60°
(3) 90°
(4) 120°
4. The locus of midpoints of the chords of the ellipse 2 2 22 1 xy ab += which passes through the foot of perpendicular on di rectrix, is
(1) 2 2 22 y xx abae +=
(2) 2 2 22 y xax abe +=
(3) 2 2 22 xye ab +=
(4) 2 2 22 y xex aba +=
5. If the product of the slopes of the tangents to the ellipse 2x2+3y2 = 6, drawn from the point (1, 2), is k, then k ____.
6. The radius of the director circle of 16x2+9y2 = 144 is ____.
Answer Key
(1) 3 (2) 1 (3) 3 (4) 1
(5) 1 (6) 5
CHAPTER REVIEW
Introduction to Ellipse
1. In a plane, the locus of a point whose distances from a fixed point and a fixed straight line are in constant ratio e , where 01 e << , is called an ellipse.
2. The equation 222220 axhxybygxfyc +++++= represents ellipse if 2 0 hab−< and 222 20abcfghafbgch ∆=+−−−≠
3. Let P be any point on the ellipse. Then, 2,SPSPa ′ += where 2a is the length of the major axis and , SS ′ are foci.
4. The angle made by the chord joining the end points of major axis and minor axis of ellipse with x-axis is 1 22 sin b ab
5. The locus of a point, the sum of whose distances from two fixed points is a constant k, provided the distance between the fixed points is less than k, is an ellipse.
6. If PSQ is a focal chord of the ellipse, then += 2 112a SPSQb .
Different Forms of Ellipse
7. Equation of ellipse in standard form 2 2 22 1, xy ab+=() 0 ab>> . This is an ellipse having major axis along the x -axis and centre at origin. This is called horizontal ellipse. Y Y’ Z B C S S’ Z’ A’ A B’
8. For the ellipse 2 2 22 1 xy ab += , () 0 ab>>
• Centre is () 0,0 C
• The major axis is along x -axis and equation of major axis is y = 0.
• The minor axis is along y -axis and equation of minor axis is x = 0.
• The length of major axis is 2 a and the length of the minor axis is 2 b.
• The ends of the major axis are vertices ()() ,0,,0AaAa ′
• The ends of the minor axis are ()() 0,,0, BbBb ′ .
• Eccentricity is 22ab e a = .
• Foci are ()() ,0,,0SaeSae ′ and the distance between the foci is 2ae .
• If P is any point on the ellipse, then 2 SPSPa ′ +=
• The length of the latusrectum is
2 2b a and the equations of the latusrecta are , xaexae ==− .
• Ends of the latus rectum are
2 , b ae a ±±
• Equations of directrices are a x e =± and distance between the directrices is 2a e
9. The ellipse 2 2 22 1 xy ab += , () 0 ab<< This is an ellipse having major axis along the y -axis and centre at origin. This is called vertical ellipse.
10. For the ellipse 2 2 22 1 xy ab += , () 0 ab<<
• Centre is () 0,0 C
• The major axis is along y -axis and equation of major axis is x = 0.
• The minor axis is along x -axis and equation of minor axis is y = 0.
• The length of major axis is 2 b and the length of the minor axis is 2a.
• The ends of the major axis are vertices ()() 0,,0, BbBb ′ .
• The ends of the minor axis are ()() ,0,,0AaAa ′ .
• Eccentricity is 22ba e b =
• Foci are ()() 0,,0, SbeSbe ′ and distance between the foci is 2 SSbe ′ = .
• If P is any point on the ellipse, then 2 SPSPb ′ += .
• The length of the latusrectum is 2 2a b and the equations of the latusrecta are , ybeybe ==− .
• Ends of the latusrecta are 2 , abe b ±±
• Equations of directrices are b y e =± .
11. Let the minor and major axes of the ellipse be along two mutually perpendicular lines 1111 0 Laxbyc≡++= and 2112 0 Lbxayc≡−+= , respectively. Let a and b be the lengths of semi-major axis and semi-minor axis, respectively, () ab >
. The equation of the ellipse is 22 111112 22 22 11 11 22 1 ++−+ ++ += axbycbxayc abab ab
Tangent and Normal
12. If 2222 camb <+ , then the line ymxc =+ intersects the ellipse 2 2 22 1 xy ab +=
13. The slope form of tangent to the ellipse 2 2 22 1 xy ab += is 222 ymxamb =±+ and its point of contact is 22 , amb cc , where 222 camb =±+ .
14. The condition for the line 0 lxmyn++= to be tangent to the ellipse 2 2 22 1 xy ab += is 22222 albmn += and the point of contact is 22 , albm nn
15. If ab > , then the equation of auxiliary circle of the ellipse 2 2 22 1 xy ab += is 222 xya +=
16. Product of the perpendiculars from foci on any tangent to the ellipse 2 2 22 1 xy ab += is 2 b . (If a>b)
17. The tangents at the extremities of the latus rectum of an ellipse intersect on corresponding directrix.
18. The portion of the tangent to an ellipse between the point of contact and the directrix subtends right angle at corresponding focus.
19. The equation of normal to the ellipse 2 2 22 1 xy ab += at () 11 , Pxy is 2 2 22 11 axbyab xy −=− .
20. The condition for the line 0 lxmyn++= to be a normal to the ellipse 2 2 22 1 xy ab += is ()2 22 22 222 abab lmn += .
21. The slope form of normal to the ellipse
2 2 22 1 xy ab += is () 22 222 mab ymx abm =± + .
22. If the normal at P meets major axis at G, then SG=e(SP), SG/S’G = SP/S’P. Let S and S ′ be two foci of an ellipse. The normal at P to the ellipse is internal angular bisector of SPS ∠′ and tangent at P to the ellipse is external angular bisector of SPS ∠′
23. Reflection property: If an incoming ray passes through one focus (S) and strikes the concave side of the ellipse, then the reflected ray passes through the other focus.
Parametric Equations
24. The parametric equations of the ellipse 2 2 22 1 xy ab += are cos,sinxaybθθ== .
25. The focal distance of a point ()() 11,cos,sinPxyab=θθ is
1 cos SPaexaae=−=−θ and
1 cos SPaexaae ′θ =+=+
26. The condition for the chord joining , αβ to be a focal chord is coscos 22 e
Pair of Tangents
29 If 12 , mm are the slopes of the two tangents drawn from a point () 11 , Pxy to the ellipse then 11 12 22 1 2 xy mm xa += and 22 1 12 22 1 yb mm xa = .
30. The locus of the point of intersection of perpendicular tangents to an ellipse is a circle. This circle is called director circle. The equation of director circle to the ellipse 2 2 22 1 xy ab += is 2222 xyab +=+ .
31. The midpoint of the chord 0 lxmyn++= of the ellipse 2 2 22 1 xy ab += is 22 22222222 ln , abmn albmalbm
32. Concyclic Points on ellipse: If a circle cuts an ellipse in four real or imaginary points, then the sum of the eccentric angles of these four concyclic points on the ellipse is an even multiple of p .
33. Co-normal points: The points on the ellipse, at which the normals drawn to the ellipse passes through a given point are called conormal points.
• At most four normals can be drawn from any point to the ellipse
27. The equation of tangent at () P θ to the ellipse
2 2 22 1 xy ab += is cossin1xy ab θθ+= .
28. The area of the ellipse
2 2 22 1 xy ab += is abπ . The maximum area of the rectangle inscribed in the above ellipse is 2ab and the sides are 2,2ab
• The sum of the eccentric angles of the feet of the normals to an ellipse through a point is an odd multiple of π .
• If 1234 ,,, θθθθ are eccentric angles of four co-normal points on the ellipse, 2 2 22 1 xy ab += , then () 12 cos0 θθ ∑+= and () 12 sin0 θθ ∑+= .
Exercises
JEE MAIN LEVEL
LEVEL-I
Different Forms of Ellipse
Single Option Correct MCQs
1. The eccentricity of an ellipse is 3 2 , then its length of latus rectum is
(1) 1 2 (length of major axis)
(2) 1 3 (length of major axis)
(3) 1 4 (length of major axis)
(4) 2 3 (length of major axis)
2. The foci of the ellipse 25 x 2 +4 y 2 +100 x –
4y+100 =0
±
(1)
(3)
521 ,2 10
221 ,2 10 ±
(2)
(4)
±
521 2, 10
221 2, 10
3. The length of the latusrectum of 9x2+25y2 –90x–150y+225 = 0 is (1) 9 5 (2) 18 5 (3) 18 25 (4) 9 25
4. The major axis and minor axis of an ellipse are respectively, x–2y–5 = 0, and 2x+y+10 = 0, one end of latus rectum is (3, 4), then foci are
(1) (5, 0), (–3, –4) (2) (5, 0), (–6, –4)
(3) (5, 0), (–11, –8) (4) (5, 0), (11, –4)
5. A focus of an ellipse is at the origin the directrix is the line x = 4 and the eccentricity is 1/2. Then the length of semi major axis is (1) 8 3 (2) 2 3 (3) 4 3 (4) 5 3
6. An ellipse having the coordinate axes as its axes and its major axis along y–axis, passes through the point (–3, 1) and has eccentricity 2 5 , then its equation is (1) 3x2+5y2–15 = 0 (2) 5x2+3y2–32 = 0 (3) 3x2+5y2–32 = 0 (4) 5x2+3y2–48 = 0
7. An ellipse passing through () 4 2,2 6 has foci at (–4, 0) and (4, 0). Its eccentricity is (1) 2 (2) 1 2 (3) 1 2 (4) 1 3
8. The eccentricity of an ellipse having centre at the origin axes along the coordinate axis and passing through the points (4, –1) and (–2, 2) is (1) 3 2 (2) 3 4 (3) 2 5 (4) 1 2
9. Let S and S' be the foci, LL ' is the latus rectum of an ellipse and S11 LL ∆ is an equilateral triangle. Then e = (1) 1 2 (2) 1 3 (3) 1 5 (4) 2 3
10. Let S and S' be the foci of an ellipse B be one end of its minor axis. If SBS' is an isosceles right angled triangle then the eccentricity of the ellipse is (1) 1 2 (2) 1 2 (3) 3 2 (4) 1 3
11. If S and S' are the foci of the ellipse 2 2 1 2516 xy+= and if PSP ' is a focal c hord with SP = 8, then SS ' = (1) 4 + S'P (2) S'P – 1 (3) 4 + SP (4) SP – 1
12. The eccentricity of the ellipse given by the locus of the point P( x , y ) satisfying the equation ()()()() 2222 21218xyxy −+−+++−= is (1) 1 8 (2) 1 4 (3) 1 2 (4) 1 2
13. If S and S' are the foci BB' is the minor axis such that 11 3 sin 5 SBS ∠= , then e = (1) 1 3 (2) 1 5 (3) 1 10 (4) 1 2
14. A circle is described with minor axis of an ellipse as a diameter. If the foci lie on the circle, then eccentricity of the ellipse is (1) 1 2 (2) 1 3 (3) 1 3 (4) 1 2
15. If a point P(x, y) moves along the ellipse 2 2 1 2516 xy+= and if C is the centre of the ellipse, then the sum of maximum and minimum values of CP is (1) 25 (2) 9 (3) 4 (4) 5
16. x2+4y2+2x+16y+k = 0 represents an ellipse with eccentricity 3 2 , for (1) only one value of k (2) only two real values of k (3) infinite number of values of k (4) no real value of k
18. Let S and S1 are the foci of an ellipse whose eccentricity is 1 2 , B and B' are the ends of minor axis then SBS B' forms a (1) Parallelogram (2) Rhombus (3) Square (4) Rectangle
19. If a and c are positive real numbers and the ellipse 2 2 22 1 4 xy cc += , has four distinct points in the common with circle x2+y2 =
17. An ellipse has OB as minor axis. F and F' are its foci and the angle FBF' is a right angle. Then the eccentricity of the ellipse is (1) 1 2 (2) 3 2 (3) 2 3 (4) 1 2
20. P is a variable point on the ellipse 2 2 22 1 xy ab += with AA' as th e major axis. Then the maximum value of the area of ∆ APA' is (1) ab (2) 2ab (3) 2 ab (4) 3 ab
21. Let S and S1 be the foci of an ellipse. At any point P on the ellipse if SPS1<90°, then its eccentricity (1) 1 e
22. Axes are co-ordinate axes, S and S 1 are Foci, B and B1 are the ends of minor axis |1 4 sin. 5 SBS
If area of SBS1B1 is 20 sq.units then equation of the ellipse is (1) 2 2 y x 1 2016 += (2) 2 2 1 2516 xy+= (3)
23. The set of values of ‘ a ’ for which (13 x –1) 2 +(13 y –2) 2 = a (5 x +12 y –1) 2 represents, an ellipse if
(1) 1 < a < 2 (2) 2 < a < 3
(3) 0 < a < 1 (4) 3 < a < 4
24. Foci of an ellipse are at S(1, 7), S' (1, –3). The point P is on the ellipse such that SP = 7, S'P = 5. Then the equation of the ellipse is
(1) ()() 22 12 1 1136 xy−+ +=
(2) ()() 22 12 1 1136 xy +=
(3) ()() 22 12 1 3611 xy +=
(4) ()() 22 12 1 116 xy +=
25. If the equation 8[(x+1)2+(y–1)2] = (x–y+3)2 represents a conic. The equation of its latus rectum is
29. An ellipse is inscribed in a rectangle and if the angle between the diagonals is 1 tan22 then e = (1)
30. The ellipse x 2 +4 y 2 = 4 is inscribed in a rectangle aligned with the coordinate axes, which in turn is inscribed in another ellipse that passes through the point (4, 0). Then, the equation of the ellipse is (1) x2+12y2 = 16 (2) 4x2+48y2 = 48 (3) 4x2+64y2 = 48 (4) x2+16y2 = 16
Numerical Value Questions
31. If the latusrectum of an ellipse is half of its major axis then 2e2 is _____.
32. If the length of the latus rectum of the ellipse x2+4y2+2x+8y–λ = 0 is 4 and l is the length of its major axis, then λ+l is equal to _____.
Tangent and Normal
Single Option Correct MCQs
33. If a tangent of slope 2 of the ellipse 2 2 22 1 xy ab += is normal to the circle x2+y2+4x+1 = 0, then the maximum value of ab is (1) 4 (2) 2 (3) 1 (4) none of these
34. If the curve 2 2 1 4 xy α += and y 2 = 16 x intersect at right angles, then a value of α is (1) 2 (2) 4 3 (3) 1 2 (4) 3 4
35. The equations of the tangents to the ellipse 4x2+3y2 = 5 which are perpendicular to the line 3x–y+7 = 0 are
(1) 2–2 55 0 xy ±=
(2) 2–12 55/20 xy ±=
(3) 26 65 0 xy+±=
(4) 22 15 0 xy+±=
36. If the tangent at the point (1, 2) on the ellipse 3x2+4y2 = 19 is also a tangent to the parabola y2 –kx = 0, then k =
(1) 57 16 (2) 57 64 (3) 57 64 (4) 57 16
37. The minimum area of the triangle formed by any tangent to the ellipse 2 2 1 1681 xy+= and the coordinate axes is (1) 12 (2) 18 (3) 26 (4) 36
38. The eccentricity of an ellipse whose centre is at the origin is 1/2. If one of its directrixes is x = 4, then the equation of the normal to it at (1, 3/2) is (1) x+2y = 4 (2) 2y–x = 2 (3) 4x–2y = 1 (4) 4x+2y = 7
39. A point on the ellipse 4x2+9y2 = 36, where the normal is parallel to the line 4 x–2y–5 = 0 is
40. The tangent at any point P on the ellipse meets the tangents at the vertices A, A 1 of the ellipse 2 2 22 1 xy ab += at L and M respectively. Then AL.A1M= (1) a2 (2) b2 (3) a2+b2 (4) ab
41. Let Pi and Pi' be the feet of the perpendiculars drawn form foci S, S' on a tangent T to an ellipse whose length of semi-major axis is 20. If () () 10 ' 1 '2560ii i SPSP = ∑= , then the value of eccentricity = (1) 1 5 (2) 2 5 (3) 3 5 (4) 4 5
Numerical Value Questions
42. If x + ky –5 = 0 is a tangent to the ellipse 4x2+9y2 = 20, then |k| = _______.
43. If a tangent to the ellipse 2 2 22 1()xyab ab +=> meets major and minor axis at M and N respectively, and C is centre of the ellipse, then ()() 22 22 ab CMCN += ______.
44. The product of the perpendicular distances from the points (3, 0) and ( –3, 0) to the tangent of the ellipse 2 2 1 3627 xy+= at 9 3, 2
is _____.
Parametric Equations
Single Option Correct MCQs
45. The distance of a point P on the ellipse 2 2 1 124 xy+= from centre is 6 , then the eccentric angle of P is (1) 2 π (2) 6 π (3) 4 π (4) 3 π
46. The coordinates of a point, in the parametric form, on the ellipse whose foci are (–1, 0) and (7, 0) and 1 2 e = are
(1) () 8cos ,43 sin θθ
(2) () 38cos ,43 sin +θθ
(3) () 343cos ,8 sin +θθ
(4) () 343cos ,23 sin +θθ
47. If the eccentric angles of the extremities of a focal chord (other than the major axis) of the ellipse 2 2 1 259 xy+= are α and β then () () cot/2 . tan/2 α β =…… (1) 4 3 (2) –9 (3) 9 (4) 4 5
48. The line 12 cos 5 sin 60 xyθθ+= is tangent to which of the following curves?
(1) 25x2+12y2 = 3600
(2) x2+y2 = 169
(3) x2+y2 = 60
(4) 144x2+25y2 = 3600
49. The line lx+my+n = 0 will cut the ellipse 2 2 22 1 xy ab += in points whose eccentric angles differs by 2 π if
(1) a2l2 + b2m2 = n2
(2) a2l2 + b2m2 = 2n2
(3) a2l2 + b2m2 = 2 2 n
(4) a2l2 + b2m2 = 4n2
50. A tangent is drawn to the ellipse 2 2 1 27 x y += at the po int () 33 , cossinθθ , where θπ<< 0 2 then the sum of the intercepts of the tangent with the coordinate axes is least when θ = (1) 6 π (2) 3 π (3) 8 π (4) 4 π
Pair of Tangents
Single Option Correct MCQs
51. Number of points on the ellipse 2 2 1 5020 xy+= from which pair of perpendicular tangents are drawn to the ellipse 2 2 1 169 xy+= is (1) 4 (2) 2 (3) 5 (4) 1
52. The sum of the slopes of the tangents to the ellipse 2 2 1 94 xy+= drawn from the point (6, –2) is (1) 0 (2) 3 4 (3) 6 7 (4) 8 9
53. The midpoint of the chord 2 x+5y = 12 of the ellipse 4x2+5y2 = 20 is (1) (6, 0) (2) (1, 2) (3) 3 ,3 2 (4) (11, 2)
54. The midpoint of the chord of the ellipse x2+4y2–2x+20y = 0 is (2, –4). The equation of the chord is (1) x–6y = 26 (2) x+6y = 26 (3) 6x–y = 26 (4) 6x+y = 26
55. A ray emnating from the point (-4, 0) is incident on the ellipse mirror 9 x 2 +25 y 2 = 225 at the point P with abscissa 3. The equation of the reflected ray after first reflection is ax+by = c (HCF of a, b, c is 1 ) then |b|is (1) 4 (2) 5 (3) 6 (4) 7
LEVEL-II
Different Forms of Ellipse
Single Option Correct MCQs
1. If the ellipse 2 2 1 4 x y += meets the ellipse 2 2 2 1 xy a += in four distinc t points a nd a = b2–5b+7, then b does not lie in (1) [4, 5] (2) ()() ,23, −∞∪∞ (3) () ,0−∞ (4) [2, 3]
2. Let a line L passes through the point of intersection of the lines bx +10 y –8 = 0 and 2 x –3 y = 0, 4 bR 3
If the line L also passes through the point (1, 1) and touches the circle 17(x2+y2) = 16, then the eccentricity of the ellipse 2 2 2 1 5 xy b += is (1)
3. The ellipse 2 2 22 1 xy ab += is such that it has the least area but contains the circle ( x –1) 2 + y 2 = 1, then the length of latus rectum of ellipse is equal to (1) 1 2 (2) 2 (3) 2 (4) 4
4. An ellipse is drawn by taking a diameter of the circle ( x –1) 2 + y 2 = 1 as its semi minor axis and a diameter of the circle x 2 +( y –2) 2 = 4 as its semi-major axis. If the centre of the ellipse is at the origin and its axes are the coordinate axes, then the equation of the ellipse is
5. (x–2) 2 +(y+3) 2 = 16 touching the ellipse ()() 22 22 23 1 xy pq −+ += from inside. If (2, –6) is one focus of t he ellipse then (p, q) = (1) (4, 5) (2) (5, 4) (3) (5, 3) (4) (3, 5)
Tangent and Normal Single Option Correct MCQs
6. Angle subtended by common tangents of two ellipses 4( x -4) 2 +25 y 2 = 100 and 4(x+1)2+y2 =4 at origin is (1) 3 π (2) 4 π (3) 6 π (4) 2 π
7. A tangent to the ellipse 2 2 22 1 xy ab += touches at the point P on it in the first quadrant and meets the coordinate axes in A and B respectively. If P divides AB in the ratio 3:1, then equation of tangent is (1) 3 axbyab += (2) 32 axbyab += (3) 3 bxayab += (4) 32 bxayab +=
8. The tangent to the ellipse 3x 2+16y 2 = 12, at the point (1, 3/4), intersects the curve y2+x = 0 at (1) no point (2) exactly one point (3) two distinct points (4) more than two points
9. P is a point on the ellipse 2 2 22 1 xy ab += with foci at S, S'. Normal at P cuts the x–axis at G and if 2 3 SP SP = ′ then SG SG ′ =
(1) 4 9 (2) 3 2 (3) 2 3 a b (4) 2 3
10. The line 2 211(1) pxypp+−=< for different values of p, touches
(1) an ellipse of eccentricity 2 3
(2) an ellipse of eccentricity 3 2
(3) hyperbola of eccentricity 2
(4) hyperbola of eccentricity 2
Parametric Equations
Single Option Correct MCQs
11. The area of the parallelogram formed by the tangents at the points whose eccentric angles 3 , , , 22 θθππ πθθ +++ on the ellipse 2 2 22 1 xy ab += is
(1) ab (2) 2ab (3) 3ab (4) 4ab
Numerical Value Questions
12. Let P be a point on an ellipse whose parameter is 3 π . The sum and differences of the focal distances of P are 8 and 3 then the eccentricity of the ellipse is _____.
13. If the minimum area of the triangle formed by a tangent to the ellipse 2 2 22 1 4 xy ba += and the coordinate axis is kab, then k is equal to ____.
14. The equation of the normal to the ellipse 2 2 1 169 xy+= at the point whose eccentric angle 8633 6 isxyk θπ=−= , then k is ___.
Pair of Tangents
Single Option Correct MCQs
15. The acute angle between the pair of tangents drawn to the ellipse 2x2+3y2 = 5 from the point (1, 3) is
(1) 1 16 tan 75
(3) 1 32 tan 75
(2) 1 tan 7 24 5
(4) 1 385 tan 35 +
16. The equations of the tangent to the ellipse 9x2+16y2 = 144 which pass through the point (2, 3) is
2 2 1 :1 49 Cxy+= and 2 2 2 :1 42143 Cxy−= does not pass through the fourth quadrant. If T touches C1 at (x1, y1) and C2 at (x2, y2), then |2x1 + x2| is equal to (1) 10 (2) 20 (3) 40 (4) 60
18. If the tangents drawn from a point P to the ellipse 22 4924360 xyxy+−+= , are perpendicular, then the locus of P, is
19. Let P1 and P2 be the feet of perpendiculars from foci of ellipse 2 2 1 2516 xy+= upon any tangent at P on the ellipse. Now tangents at P1 and P2 to the auxilliary circle meets at Q. Then eccentricity of the locus of the point Q is (1) 1 (2) 0.8 (3) 2 (4) 0.6
20. The shortest distance between the ellipse x2+2 y2 = 2 and the circle x2+y2–3x–2 2 y+4 = 0 is (1) 31 2 + (2) 3 +1 (3) 3 –1 (4) 31 2
21. A ray of light along the line x = 3 is reflected at the ellipse 2 2 1 2516 xy+= . The slope of the reflected ray is (1) 4 15 (2) 8 15 (3) 15 8 (4) 15 4
Numerical Value Questions
22. The number of tangents to 2 2 1 259 xy+= through (1, 1) is _____.
23. Let the tangents at the points P and Q on the ellipse 2 2 1 24 xy+= meet at the point () 2, 222. R If S is the focus of the ellipse on its negative major axis, then SP2+SQ2 is equal to _____.
LEVEL-III
Single Option Correct MCQs
1. If P is a point on the ellipse of eccentricity e and A, A' are the vertices and S, S' are the foci then area of ∆ SPS' : area of ∆ APA' = (1) e3 (2) e2 (3) e (4) 1 e
2. Let PQ be a focal chord of the parabola y 2 = 4 x such that it subtends an angle of 2 π at the point (3, 0). Let th e line segment PQ be also a focal chord of the ellipse 2 2 22 22 :1, xy Eab ab +=> . If e is the eccentricity of the ellipse E, then the value of 2 1 e is equal to
(1) 12 + (2) 322 + (3) 123 + (4) 453 +
3. If the ellipse 2 2 22 1 xy ab += meets the line 1
7 26 xy+= on the x–axis and the line 1 7 26 xy −= on the y-axis, then the eccentricity of the ellipse is
(1) 5 7 (2) 26 7 (3) 3 7 (4) 25 7
4. Let E be an ellipse whose axes are parallel to the co-ordinate axes, having its centre at (3, –4) one focus at (4, – 4) and one vertex at (5, -4). If mx – y = 4, m > 0 is a tangent to the ellipse E, then the value of 5 m 2 is equal to (1) 1 (2) 2 (3) 3 (4) 4
5. The point P 4 π lie on the ellipse 2 2 1 42 xy+= whose foci S and S 1 . The equation of the external angular bisector of |SPS of ∆ le SPS1 is
(1) x+ 2 y = 2 2
(2) 2x+3 3 y = 12
(3) 3x–4y+12 2 = 0
(4) x+y+12 2 = 0
6. An ellipse has the points (1, –1) and (2, –1) as its foci and x + y –5 = 0 as one of its tangents. Then the point where this line touches (1) 3222 , 99
(2) 232 , 99
(3) 3411 , 99 (4) (1, 1)
7. If the tangents at the point 16 4cos2,sin2 11 θθ
on the elli pse 16x2+17y2 = 256 touches the circle x2+y2–2x = 15, then θ = (1) 3 π ± (2) 6 π ± (3) 4 π ± (4) 8 π ±
8. Tangents are drawn to the ellipse 2 2 22 xyab ab +=+ at the points where it is cut by the line 22cossin1, xy ab θθ−= then the point of intersection of Tangents
(1) ()() () cos,sinababθθ+−+
(2) ()() () cos,sinababθθ
(3) ()() () sin,cosababθθ+−+
(4) ()() () cos,sinababθθ+−−
9. Tangents to the ellipse 2 2 22 1 xy ab += makes angles θ1 and θ2 with the major axis of the ellipse such that ( θ 1+ θ 2) = k (constant). The locus of the point of intersection of the tangents is (x2 –a2)–(y2 –b2) =
(1) 2 k xy (2) 2 kxy
(3) xy k (4) kxy
10. Locus of the point of intersection of the tangents at the points with eccentric angle
θ and 2 πθ + is
(1) x2+y2 = a2 (2) x2+y2 = b2
(3) 2 2 22 2 xy ab += (4) 2 2 22 1 xy ab +=
11. A ray of light through (2, 1) is reflected at a point P on the y axis and then passes through the point (5, 3). if this reflected ray is the directrix of an ellipse with eccentricity 1 3 and the distance of the near focus from the directrix 16 53 Then the equation of the directrix can be
12. If the normal at one end of latus rectum of an ellipse 2 2 22 1 xy ab += passes through one end of minor axis, then e4+e2 = ______.
13. If two tangents drawn from a point ( α, β) lying on the ellipse 25 x 2 + 4 y 2 = 1 to the parabola y2 = 4x are such that the slope of one tangent is four times the other, then the value of (10 α +5) 2+(16 β 2+50) 2 equals _____.
THEORY-BASED QUESTIONS
Very Short Answer Questions
1. If the eccentricity is less than 1, then what is the name of the conic?
2. For any point P on the ellipse, having foci , SS ′ then what is the value of SPSP +′ ?
3. What is the name of the line segment passing through the foci of the ellipse and end points on the ellipse?
14. Let the common tangents to the curves 4(x2+y2) = 9 and y2 = 4x intersect at the point Q. Let an ellipse, centered at the origin O, has lengths of semi-minor and semi-major axes equal to OQ and 6 , respectively. If e and l respectively denote the eccentricity and the length of the latus rectum of this ellipse, then 2 e l is equal to _____.
15. The line passing through the extremity A of the major axis and extremity B of the minor axis of the ellipse x2+9y2 = 9, meets its auxiliary circle at the point M. Then the area of the triangle with vertices at A, M and the origin O is _____.
16. The value of ‘ a ’ for the ellipse 2 2 22 1,()xyab ab +=> if the extremities of the latusrectum of the ellipse having positive ordinates lie on the parabola x2 = –2(y-2) is_____.
17. P is a point on ellipse 2 2 1 2516 xy+= with foci S and S1. The maximum area of ∆SPS1 is _____.
18. The product of perpendiculars from foci of 2 2 1 2516 xy+= to any tangent is ______.
19. If 4x2+y2 = 1, then maximum value of 12x2-3 y2+16xy is (x, y ∈ R) _____.
4. If , ab are the lengths of semi major axis and semi minor axis of ellipse, then write the formula for eccentricity.
5. What is the length of the latusrectum of ellipse 2 2 22 1 xy ab += when ba > ?
6. What is the equation of tangent to the ellipse 2 2 22 1 xy ab += at a point whose eccentric angle is α ?
7. If the distance between a point P on the ellipse from major axis is 4 units and the distance between corresponding point of P and major axis is 5 units then what is the eccentricity ?
8. What is the equation of normal at θ to the ellipse 2 2 22 1? xy ab +=
9. If 12 , mm are the slopes of two tangents drawn from a point which lies on the director circle of the ellipse then what is the value of m1.m2?
10. Suppose that the tangent at P to the ellipse intersect the directrix at T . If S is corresponding focus, then find orthocentre of the triangle STP
Statement Type Questions
Each question has two statements: statement I (S-I)and statement II (S-II). Mark the correct answer as
(1) if both statement I and statement II are correct
(2) if both statement I and statement II are incorrect
(3) if statement I is correct, but statement II is incorrect
(4) if statement I is incorrect, but statement II is correct
11. S-I : A chord passing through either of the foci is called focal chord.
S-II : A focal chord of an ellipse perpendicular to the major axis is called latusrectum.
12. S-I : If PSQ is a focal chord then ,, SPSLSQ are in harmonic progression, where SL is semi latus rectum.
S-II : If PSQ is a focal chord, then 2 11 a SPSQb +=
13. S-I : The equations of directrixes to the ellipse 2 2 22 1; xyab ab +=> are a x e =± .
S-II : The distance between the two directrixes of the ellipse 2 2 22 1; xyab ab +=> is 2 22 2a ab
14. S-I : For the ellipse 2 2 22 1; xyba ab +=> , let P be any point on the ellipse then 2 SPSPb ′ +=
S-II : For the ellipse 2 2 22 1; xyba ab +=> , suppose that PSQ is a focal chord, then 2 112a SPSQb +=
15. S-I : The line ymxc =+ is tangent to the ellipse 2 2 22 1 xy ab += , if 2222 camb =+
S-II : The slope form of a tangent to the ellipse is 222 ymxamb =±+
16. S-I : Let () 11 , Pxy be any point in the plane of ellipse 0 S = then the equation 1 0 S = is tangent when () 11 , Pxy lies on the ellipse.
S-II : Let () 11 , Pxy be any point in the plane of ellipse 0 S = then the equation 1 0 S = is chord of contact when () 11 , Pxy lies inside the ellipse.
17. S-I : The tangents at the extremities of latusrectum of an ellipse intersect on corresponding directrix.
S-II : The tangents at extremities of focal chord of an ellipse meet on the corresponding directrix.
18. S-I : The product of the perpendiculars from the foci of any tangent to an ellipse is equal to the square of the semi minor axis
S-II : The feet of the perpendiculars drawn from foci to any tangent lies on the auxiliary circle.
19. S-I : The equation of tangent to the ellipse 2 2 22 1 xy ab += at θ is
cossin1xy ab θθ+=
S-II : The equation of normal to the ellipse at θ is 22 cossin axbyab θθ −=−
20. S-I : The maximum area of the rectangle inscribed in the ellipse 2 2 22 1 xy ab += is 2ab
S-II : The sides of the rectangle of maximum area that inscribed in the ellipse are 2,2ab
Assertion and Reason Questions
In each of the following questions, a statement of Assertion (A) is given, followed by a corresponding statement of Reason (R). Mark the correct answer as
(1) if both (A) and (R) are true and (R) is the correct explanation of (A)
(2) if both (A) and (R) are true but (R) is not the correct explanation of (A)
(3) if (A) is true but (R) is false
24. (A) : The equation of director circle of the ellipse ()() 22 22 1; xhykab ab +=> is ()() 22 22xhykab −+−=+
(R) : The angle between two tangents drawn from any point on ()() 22 2 xhyka −+−= to the ellipse ()() 22 22 1; xhykab ab +=> is right angle.
25. (A) : Ratio of area of triangle PQR inscribed in an ellipse and area of triangle formed by corresponding points of vertices of PQR is constant.
(R) : The corresponding point of () cos,sinPabθθ is () cos,sinPaa′θθ
26. (A) : The centre of the ellipse ()() 22 111111 22 11 22 ; axbycbaycabab ab ++−+ +=+>
()() 22 111111 22 11 22 ; axbycbaycabab ab ++−+ +=+> is the point of intersection of 111 0 axbyc++= and 112 0 bxayc−+=
(4) if both (A) and (R) are false
21. (A) : The major axis of the ellipse 2 2 22 1; xyab ab +=> is 0 y =
(R) : The major axis is a line segment passing through foci and having end points on the curve.
(R) : The equations of major axis and minor axis of ()() 22 111111 22 11 22 ; axbycbaycabab ab ++−+ +=+>
()() 22 111111 22 11 22 ; axbycbaycabab ab ++−+ +=+> are 111 0 axbyc++= and 112 0 bxayc−+= .
22. (A) : The length of the latus rectum to the ellipse 2 2 22 1; xyab ab +=> is 2 2b a
(R) : Latus rectum is a focal chord which is perpendicular to the major axis.
23. (A) : The line ymxc =+ is a tangent to the ellipse 2 2 22 1 xy ab += when 2222 camb =+
(R) : The slope form of tangent to the ellipse 2 2 22 1 xy ab += is 222 ymxamb =±+
27. (A) : The focal distance of a point θ is cos aae θ
(R) : The focal distance of () 11 , Pxy is 1aex
28. (A) : The circle on any focal distance of ellipse as diameter touches the auxiliary circle.
(R) : Two circles having centre at 12 , CC and radii 12 , rr are touch each other externally then 1212 CCrr =+ .
29. (A) : If tangent at point P on the ellipse intersect major axis at T. If N is foot of perpendicular from P on major axis. Then circle drawn on NT as diameter intersect the auxiliary circle orthogonally.
(R) : If any tangent to an ellipse intersect the tangents at the vertices at T and T’. Then the circle considering TT’ as diameter passess through both foci.
JEE ADVANCE LEVEL
Multiple Option Correct MCQs
1. If two concentric ellipses be such that the foci of one be on the other and their major axes are equal. If e1 and e2 be their eccentricities, then the angle θ between their axes is given by:
(1) 2222 1212 111 cos eeee
(2) 2222 1212 111 cos eeee θ =++
(3) 2 2 2222 1212 111 cos eeee
(4) 22 12 11 sin11
2. If equation of the ellipse is 3x2+2y2+6x–8y+5 = 0, then which of the following are true:
(1) 1 3 e =
(2) centre is (–1, 2)
(3) foci are (–1, 1) and (–1, 3)
(4) directrices are 23 y =±
3. If P is any point lying on the ellipse 2 2 22 1 xy ab += whose foci are S and S’. Let
PSS α ′ ∠= and PSS β ′ ∠= then which of the following is/are true:
(1) PS+PS’ = 2a, if a > b
(2) PS+PS’ = 2b, if a < b
(3) 1 tantan 221 e e αβ = +
4. If pair of tangents drawn to the ellipse 2 2 1 169 xy+= from a point P so that angle between the tangents is a right angle, then possible coordinates of the point P is/are: (1) (3, 4) (2) (5, 0)
(3) () 25,5 (4) () 32,7
5. The locus of extremities of the latusrectum of the family of ellipse b 2x 2+ a 2y 2 = a 2b 2 is () bR ∈
(1) x2 –ay = a2 (2) x2 –ay = b2
(3) x2+ay = a2 (4) x2+ay = b2
6. For a point P on ellipse the circles with PS and PS' where S, S' represent foci of ellipse as diameter intersect the auxiliary circle of ellipse at A, A1 and B, B1 respectively, then which of the following is/are correct?
(1) A and A1 coincide, B and B1 coincide (2) Segment AB is tangent to ellipse at P
(3) Tangents at A and B on auxiliary circles are perpendicular
(4) SA and S’B are parallel
7. If a pair of variable straight lines x2+4y2+αxy = 0 (where α is a real parameter) cut the ellipse x2+4y2 = 4 at two points A and B, then the locus of the point of intersection of tangents at A and B is
(1) x–2y = 0 (2) 2x–y = 0
(3) x+2y = 0 (4) 2x+y = 0
8. Extremities of the latus rectum of the ellipses 2 2 22 1()xyab ab +=> having a agiven major axis 2a lies on
(1) x2 = a(a–y) (2) x2 = a(a+y)
(3) y2 = a(a+x) (4) y2 = a(a–x)
9. x–2y+4 = 0 is a common tangent to y2 = 4x and 2 2 2 1 4 xy b += Then the value of b and the other common tangent are given by:
(1) 3 b = (2) x+2y+4 = 0
(3) b = 3 (4) x–2y–4 = 0
10. The equation 2 x 2 +3 y 2 –8 x –18 y +35 = K represents (1) a point, if K = 0 (2) an ellipse, if K < 0 (3) an ellipse, if K > 0 (4) a hyperbola if K > 0
11. If the tangent to the ellipse x2+4y2 = 16 at the point P( θ ) is a normal to circle x2+y2 –8x–4y=0, then θ equals (1) 2 π (2) 4 π (3) 0 (4) 4 π
12. If the tangent at the point 16 4cos,sin 11 φφ
the the ellipse 16 x 2 +11 y 2 = 256 is also a tangent to the circle x 2+ y 2–2 x = 15, then the value of ϕ is
(1) 2 π ± (2) 4 π ± (3) 3 π (4) 3 π
13. If the normal at any given point P on the ellipse 2 2 22 1 xy ab += meets its auxiliary circle at Q and R such that ∠ QOR = 90°, where 0 is the centre of ellipse, then
(1) a4+2b4 ≥ 3a2b2
(2) a4+2b4 ≥ 5a2b2+2a3b
(3) a4+2b4 ≥ 3a2b2+ab
(4) none of these
14. (2, 6)and(12, k )are the foci of an ellipse which touches both the coordinate axes. Then its
(1) Point of contact with x-axis is (8, 0)
(2) Point of contact with y - axis is 40 0, 7
(3) Eccentricity of the ellipse is 13 5
(4) Length of latusrectum of the ellipse is 242 5
15. If the line 33 yx−= and x -2 = 0 are tangents to the ellipse 2 2 22 1 xy ab += , then
(1) Eccentricity of the ellipse is 17 23
(2) Eccentricity of the ellipse is 3 7
(3) Length of Latusrectum of the ellipse is 5 3
(4) Area of the ellipse is 5 2 3 π sq. units
Numerical/Integer Value Questions
16. P be a point on the ellipse 2 2 1 2516 xy+= with foci S and S’ if A be the area of SPS ′ ∆ then maximum value of 6 7 A is ______.
17. The normal inclined at 45° to the x–axis of the ellipse 2 2 22 1 xy ab += drawn and meets the major and minor axis in P and Q respectively. If C is the centre of the ellipse, the area of CPQ∆ is () () 2 22 22 ab Kab + square units. Then value of 4 7 K = _______.
18. A circle concentric to the ellipse
2 2 2 4 17 1 289 2 xy λ λ +=< passes throgh foci S1, S2 and cuts ellipse at point P. If area of ∆ PS 1 S 2 is 30 sq.units then find = 12 26 SS ______.
19. If F1 and F2 are the feet of perpendiculars from foci S1 and S2 of an ellipse
2 2 1 53 xy+= on the tangent at any point P on the ellipse then (S1F1)(S2F2) = ______.
20. Number of points on the ellipse
2 2 1 5020 xy+= from which pair of perpendicular tangents are drawn to the ellipse
2 2 1 169 xy+= is _______.
21. If the line lx+my+n = 0 intersects the ellipse
2 2 2 1 25 xy a += at points whose eccentric angles differ by 2 π then the value of
222 2 25 alm n + is ______.
Passage-based Questions
(Q. 22 – 23)
To prove geometrical properties of ellipse we may take its standard equation as 2 2 22 1 xy ab += We will assume that a > b > 0 and foci S, S' lie on positive and negative side of x–axis respectively.
22. If the normal at any point P meets major and minor axis at G and G' respectively and OF be the perpendicular drawn from centre O to this normal then PF. PG must be equal to (1) b2 (2) a2 (3) ab (4) a2b2
23. If P is any point on the ellipse and G is same as above, then ' SG SG must be
(1) ' OS OS (2) ' GS OG
(3) ' SP SP (4) a b
(Q. 24 – 26)
A coplanar beam of light emerging from a point source has the equation λx–y+2(1+λ)=0, λ ∈ R, the rays of the beam strike an elliptical surface and get reflected. The reflected rays form another convergent beam having equation µ x–y+2(1–µ )=0, µ∈ R. Further it is found that the foot of the perpendicular from the point(2, 2) upon any tangent to the ellipse lies on the circle x2+y2–4y–5=0.
24. The eccentricity of the ellipse is equal to (1) 1 3 (2) 1 3 (3) 2 3 (4) 1 2
25. The area of the largest triangle that an incident ray and the corresponding reflected ray can enclose with axis of the ellipse is equal to
(1) 45
(2) 25
(3) 5
(4) none of these
26. Length of the latus rectum of the ellipse is (1) 15 3 (2) 5 3
(3) 10 3 (4) None of these
(Q. 27 – 29)
Consider an ellipse 2 2 :1 415 Exy+= and a parabola P: y2 = 8x
27. Equation of a tangent common to both the parabola P and the ellipse E is
(1) x–2y+8 = 0 (2) x+2y+8 = 0
(3) x+2y–8 = 0 (4) x–2y–8 = 0
28. Equation of the normal at the point of contact of the common tangent, which makes an acute angle with the positive direction of x–axis, to the parabola P is
(1) 2x+y = 24
(2) 2x+y+24 = 0
(3) 2x+y = 48
(4) 2x+y+48 = 0
29. Point of contact of a common tangent to P and E on the ellipse is
(1)
115 , 24
(2)
115 , 24
(3) 115 , 24
(4) 115 , 24
(Q. 30 – 32)
C: x2+y2 = 9; 2 2 y x E:1, 94 += L: y = 2x
30. P is a point on the circle C, the perpendicular PL to the major axis of the ellipse E meets the ellipse at M, then ML PL is equal to
(1) 1 3 (2) 2 3 (3) 1 2 (4) 1
31. If L represents the line joining the point P on C to its centre O, then equation of the tangent at M to the ellipse E is
(1) 335xy+=
(2) 435 xy+=
(3) 3350xy++= (4) 4350 x ++=
32. If R is the point of intersection of the line L with the line x = 1, then (1) R lies inside both C and E
(2) R lies outside both C and E
(3) R lies on both C and E
(4) R lies inside C but outside E
Matrix Matching Questions
33. Match the items of List–I with the items of List–II and choose the correct option.
List - I List - II
(A) S and S' are foci and B is end of minor axis of an ellipse. If ∆ SS’B is equilateral, then eccentricity of ellipse is (p) 10
(B) If P(x, y)be a point on the ellipse 16 x2+25y2= 400 and F1 = (3, 0), F2 = (-3, 0) then PF1+PF2 = (q)
2 (3) (2) 1 43 xy + += 2 2 (3) (2) 1 43 xy + +=
(C) A circle with centre at (0, 3) passes through the foci of ellipse 2 2 1. 169 xy+= Its radius is of length (r) 1 2
(D) In an ellipse C=(2, –3), S=(3, -3) and A is (4, –3), then the equation of ellipse is (s) 4
(A) (B) (C) (D)
(1) r p s q
(2) p q r s
(3) s p r q
(4) q s r p
34. Match the following loci for the ellipse 2 2 22 1():xyab ab +=>
List - I
List - II
(A) Locus of point of intersection of two perpendicular tangents (p) (x2+y2)2 = a2x2+b2y2
(B) Locus of loot of perpendicular from any focus upon any tangent (q) (x2+y2)2 = a2x2+b2y2
(C) Locus of foot of the perpendicular from centre on any tangent (r) x2+y2 = a2
(D) Locus of midpoint of segment OM where M is the foot of the perpendicular from centre O to any tangent (s) x2+y2 = a2+b2
(A) (B) (C) (D)
(1) s r p q
(2) p q r s
(3) r s p q
(4) q r s p
35. Observe the following lists for the ellipse 16x2+9y2 = 144
List - I
List - II
(A) eccentricity (p) () 0,7 ±
(B) foci (q) 16 y 7 =±
(C) latusrectum (r) () 7,0 ±
(D) equation of directrices (s) 7 4 (t) 9 2
(A) (B) (C) (D)
(1) s p t q
(2) p q r s
(3) s t p q
(4) t s p q
36. The tangents drawn from a point P to the ellipse 2 2 22 1 xy ab += make angle α and β with the major axis
List - I List - II
(A) If () 2 c cN αβπ+=∈ , then the locus of P can be (p) Circle
(B) If tanα tanβ = c {where c ∈ R}, then locus of P can be (q) Ellipse
(C) If tanα+tanβ = e {where c ∈ R}, then locus of P can be (r) Hyperbola
(D) If cotα+cotβ = c {where c ∈ R}, then locus of P can be (s) Pair of straight lines
(A) (B) (C) (D)
(1) r, s p, q, r, s r, s r, s
(2) q p, r p r
(3) s, r r p, q p, q, r, s (4) p r, s p, q q, r, s
37. Match the items of List–I with the items of List–II and choose the correct option.
List - I List - II
(A) If the tangent to the ellipse x2+4y2 = 16 at the point P( φ ) is a normal to the curve x2+y2–8x–4y=0, then 2 φ may be (p) 0
(B) The eccentric angle(s) of a point on the ellipse x2+3y2 = 6 at a distance 2 units from the centre of the ellipse is/are (q) 1 2 cos 3
(C) The eccentric angle of intersection of the ellipse x2+4y2 = 4 and the parabola x2+1=y is (r) 4 π
(D) If the normal at the point P(θ) to the ellipse 2 2 1 145 xy+= intersects it again at the point Q(2θ), then θ is
(s) 5 4 π T) 2 π
(A) (B) (C) (D)
(1) p q r s
(2) s r q p
(3) p r q s
(4) p, r r, s t q
38. Match the items of List–I with the items of List–II and choose the correct option.
List - I List - II
(A) The number of rational points on the ellipse 2 2 1 94 xy+= is (p) Infinite
(B) The number of integral points on the ellipse 2 2 1 94 xy+= is (q) 4
(C) The number of rational points on the ellipse 2 2 x y1 3 += (r) 0
BRAIN TEASERS
(D) The number of integral points on the ellipse 2 2 x y1 3 += (s) 2
(A) (B) (C) (D)
(1) p q p s
(2) q p p r
(3) s r p q
(4) p r p q
39. Match the intercepts made by the line y = x given in List-I with the curves in List - II.
List - I List - II (A) 42 (p) 2 2 1 3 x y +=
(B) 4 (q) 2 2 1 3 xy−= (C) 6 (r) x2+y2 = 4
(D) 23 (s) y2 = 4x (t) x2 = 4y
(A) (B) (C) (D)
(1) s, t r p q
(2) p q r s
(3) t p r s
(4) r s p q
1. Suppose points F 1 , F 2 are the left and right foci of the ellipse 2 2 1 164 xy+= respectively, and point P is on line :38230lxy−++= When 12FPF∠ reaches the maximum, then the value of ratio 1 2 PF PF is (1) 31 + (2) 31 (3) 31 31 + (4) 31 31 +
2. If the equation of the curve on reflection of the ellipse 2 2 (3) (4) 1 169 xy+= about the line x–y–2 = 0 is 16x2+9y2+k1x–36y+k2 = 0, then 12 22 kk + is ______.
3. A tangent to the ellipse 2 2 1 94 xy+= (except at its vertices) meets its director circle at P and Q, then the product of the slopes of CP and CQ, where C is the centre of ellipse, is (1) 4 9 (2) 9 4 (3) 2 9 (4) depends on location of tangent
4. Let P1 and P2 be the feet of perpendiculars from foci of ellipse 2 2 1 2516 xy+= upon any tangent at P on the ellipse. Now tangents at P1 and P2 to the auxilliary circle meets at Q. Then eccentricity of the locus of the point Q is (1) 1 (2) 0.8 (3) 2 (4) 0.6
FLASHBACK (P revious JEE Q uestions
)
1. The length of the chord of the ellipse
whose mid point is
is equal to (27th Jan 2024 Shift 1)
5. Given a conic (3x+9)2+(3y–12)2–(2x–y)2 = 6y–12x+9. The eccentricity is a b (GCD (a, b) = 1) then a2+b is equal to _______.
6. An ellipse of major and minor axes of length 3 and 1 respectively, slides along the coordinate axes and always remains confined in the first quadrant. The locus of the centre of the ellipse will be the arc of a circle. The length of this arc is (1) π (2) 2 π (3) 3 π (4) 6 π
7. Consider an ellipse and a concentric circle. The circle passes through the foci of the ellipse and intersects the ellipse in four distinct points. The length of major axis of the ellipse is 15 units. If S 1 and S 2 are the foci of the ellipse and area of triangle PS1S2 is 26 sq. units, then eccentricity of the ellipse is equal to (where P is one of points of intersection of ellipse and circle) (1) 2 3 (2) 7 15 (3) 13 15 (4) 11 15
2. If the length of the minor axis of ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is (30th Jan 2024 Shift 1)
3. Let ( a , 0) and B(0, b ) be the points on the line 5x+7y = 50. Let the point P divide the line segment AB internally in the ratio 7: 3. Let 3x–25 = 0 be a directrix of the ellipse
22 :1 xy E ab += and the corresponding focus be S. If from S, the perpendicular on the x-axis passes through P, then the length of the latus rectum of E, is equal to (30 th Jan 2024 Shift 2)
4. Let P be a parabola with vertex (2, 3) and directrix 2 x + y = 6. Let an ellipse 22 22 :1, xy E ab += a > b of eccentricity 1 2 pass through the focus of the parabola P. Then the square of the length of the latus rectum of E, is (31 st Jan 2024 Shift 2)
(1) 385 8 (2) 347 8
(3) 512 25 (4) 656 25
5. Let 22 22 1, xy ab += a > b be an ellipse, whose eccentri ci ty is 1 2 and the length of the latus rectum is 14 . Then the sq uare of the eccentricity of 22 22 1 xy ab −= is:
(1st Feb 2024 Shift 1)
(1) 3 (2) 7 2 (3) 3 2 (4) 5 2
6. Let P be a point on the ellipse 22 1 94 xy+= Let the line passing through P and parallel to y-axis meet the circle x2+y2 = 9 at point Q such that P and Q are on the same side of the x-axis. Then, the eccentricity of the locus of the point R on PQ such that PR : RQ = 4 : 3 as P moves on the ellipse, is (1 st Feb 2024 Shift 2)
(1) 11 19 (2) 13 21 (3) 139 23 (4) 13 7
7. Let the ellipse E: x 2+9 y 2 = 9 intersect the positive x and y–axes at the points A and B respectively. Let the major axis of E be a diameter of the circle C. Let the line passing through A and B meet the circle C at the point P. If the area of the triangle with vertices A, P and the origin O is m n , where m and n are coprime, then m–n is equal to
(1) 17 (2) 15 (3) 16 (4) 18
8. Let a circle of radius 4 be concentric to the ellipse 15x2+19y2 = 285. Then the common tangents are inclined to the minor axis of the ellipse at the angle (1) 4 π (2) 3 π (3) 6 π (4) 12 π
9. Consider ellipse Ek: kx2+k2y2 = 1, k = 1, 2, ..., 20. Let Ck be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse E k. If r k is the radius of the circle Ck, then the value of 20 2 1 1 kk r = ∑ is (1) 3080 (2) 3320 (3) 3210 (4) 2870
10. If the radius of the largest circle with centre (2, 0) inscribed in the ellipse x2+4y2 = 36 is r, then 12r2 is equal to (1) 69 (2) 115 (3) 72 (4) 92
11. Let 236 ,, 77 P
Q, R and S be four points on the ellipse 9x2+4y2 = 36. Let PQ and RS be mutually perpendicular and pass through the origin. If ()() 22 11 , p PQRSq += , where p and q are co prime, then p+q is equal to (1) 147 (2) 143 (3) 137 (4) 157
12. Let the tangent and normal at the point () 33,1 on the ellipse 2 2 1 364 xy+= meet the y–axis at the points A and B respectively. Let the circle C be drawn taking AB as a diameter and the line 25 x = intersect C at the points P and Q. If the tangents at the points P and Q on the circle intersect at the point (),,αβ then 22αβ is equal to (1) 60 (2) 304 5 (3) 61 (4) 314 5
13. If the maximum distance of normal to the ellipse 2 2 2 1,2 4 xyb b +=< from the origin is 1, then the eccentricity of the ellipse is
(1) 1 2 (2) 3 2 (3) 1 2 (4) 3 4
14. Let C be the largest circle centered at (2, 0) and inscribed in the ellipse 2 2 1 3616 xy+= If (1, α) lies on C, then 10α2 is equal to_______
15. Let a tangent to the curve 9 x2 + 16y2 = 144 intersect the coordinate axes at the points A and B. Then, the minimum length of the line segment AB is ____.
16. The line x = 8 is the directrix of the ellipse 2 2 22 :1Exy ab += , with the corresponding focus (2, 0). If the tangent to E at the point P in the first quadrant passes through the point () 0,43 and intersects the x–axis at Q, then (3PQ)2 is equal to ______.
17. Let the maximum area of the triangle that can be inscribed in the ellipse 2 2 2 1,2 4 xya a +=> having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the y-axis, be 63 Then the eccentricity of the ellipse is:
(1) 3 2 (2) 1 2
(3) 1 2 (4) 3 4
18. The line y = x +1 meet the ellipse 2 2 1 42 xy+= at two points P and Q. If r is the radius of the circle with PQ as diameter then(3r)2 is equal to (1) 20 (2) 12 (3) 11 (4) 8
19. If m is the slope of a common tangent to the curves 2 2 1 169 xy+= and x2+y2 = 12, then 12m2 is equal to:
(1) 6 (2) 9
(3) 10 (4) 12
20. The locus of the midpoint of the line segment joining the point (4, 3)and the points on the ellipse x2+2y2 = 4 is an ellipse with eccentricity
(1) 3 2 (2) 1 22
(3) 1 2 (4) 1 2
21. Let the eccentricity of an ellipse 2 2 22 1, xyab ab +=> be 1 4 If this ellipse passes through the point 2 4, 3 5
, then a2+b2 is equal to _____.
(1) 29 (2) 31
(3) 32 (4) 34
22. Let T 1 and T 2 be two distinct common tangents to the ellipse 2 2 :1 63 Exy+= and the parabola P: y2 = 12x. Suppose that the tangent T1 touches P and E at the points A1 and A2 respectively and the tangent T 2 touches P and E at the points A4 and A3, respectively. Then which of the following statements is(are) true?
(1) The area of the quadrilateral A1A2A3A4 is 35 square units
(2) The area of the quadrilateral A1A2A3A4 is 36 square units
(3) The tangents T1 and T2 meet the x-axis at the point(-3, 0)
(4) The tangents T1 and T2 meet the x-axis at the point (-6, 0)
23. Consider the ellipse 2 2 1 43 xy+= . Let H(α, 0), 0<α<2, be a point. A straight line drawn through H parallel to the y–axis crosses the ellipse and its auxiliary circle at points E and F respectively, in the first quadrant. The tangent to the ellipse at the point E intersects the positive x-axis at a point G. Suppose the straight line joining F and the origin makes an angle φ with the positive x-axis.
List - I
(A)
(B)
If 4 φπ = , then the area of the triangle FGH is (P)
List - II
If φπ = 3 , then the area of the triangle FGH is (Q) 1
(C) If 6 φπ = , then the area of the triangle FGH is (R) 3 4
(D)
If 12φπ = , then the area of the triangle FGH is (S) 1
(1) A-R, B-S, C-Q, D-P
(2) A-R, B-T, C-S, D-P
(3) A-Q, B-T, C-S, D-P
(4) A-Q, B-S, C-Q, D-P
24. Let E be the ellipse 2 2 1 169 xy+= . For any three distinct points P, Q and Q' on E, let M(P, Q)be the midpoint of the line
segment joining P and Q, and M(P, Q')be the midpoint of the line segment joining P and Q’. Then the maximum possible value of the distance between M(P, Q)and M(P, Q’), as P, Q and Q' vary on E, is ______.
25. Let a , b and λ be positive real numbers. Suppose P is an end point of the latus rectum of the parabola y 2 = 4 λx , and suppose the ellipse 2 2 22 1 xy ab += passes through the point P. If the tangents to the parabola and the ellipse at the point P are perpendicular to each other, then the eccentricity of the ellipse is___.
(1) 1 2 (2) 1 2 (3) 1 3 (4) 2 5
26. Define the collections {E 1, E2, E3, ......}, of ellipses and {R1, R2, R3, ....} of rectangles as follows: 2 2 1 :1; 94 Exy+= R1 : Rectangle of largest area, with sides parallel to the axes, inscribed in E 1 ; E n : Ellipse 2 2 22 1 nn xy ab +=
of largest area inscribed in R n–1, n >1; R n : Rectangle of largest area, with sides parallel to the axes, inscribed in En, n>1. Then which of the following option(s) is/are correct?
(1) The eccentricities of E 18 and E 19 are NOT equal.
(2) () 1 24 N n n areaofR = ∑< , for each positive integer N
(3) The length of latus rectum of E 9 is 1 6
(4) The distance of a focus from the centre in E g is 5 32
CHAPTER TEST – JEE MAIN
Section – A
1. Let the line y = mx and the ellipse 4x2+9y2 = 1 intersect at point P in the first quadrant. If the normal to the ellipse at P meets the coordinate axes at A(α, 0) and 5 0, 83 B , then area of triangle OAB is equal to (where O is origin)
(1) 1 163 (2) 25 5763 (3) 9 1443 (4) 25 2883
2. If ()() 2 2 2 1 4 45 xy fafa += represents an ellipse with major axis as y – axis and f is decreasing function, then (1) a ∈ (– ∞ , 1) (2) a ∈ (5, ∞ ) (3) a ∈ (1, 5 ) (4) a ∈ (–1, 5 )
5. The centre of the ellipse 22 (2)() 1, 916 xyxy +−− += is (1) (0, 0) (2) (1, 1) (3) (1, 0) (4) (0, 1)
6. If the normal at any point P on the ellipse
2 2 22 1 xy ab += meets the axis of x at G and the axis of y at g, then PG : pg is : (1) a2 : b2 (2) a : b (3) b : a (4) b2 : a2
7. If F1 and F2 are the foci of the ellipse 4x2+ 9y2 = 36, P is a point on the ellipse such that PF1 : PF2 = 2 : 1; then area of ∆ PF1F2 is (1) 1 sq. unit (2) 2 sq. unit (3) 3 sq. unit (4) 4 sq. unit
8. If m is slope of common tangent to 2 2 1 169 xy+= and x 2 + y 2 = 12. Then 12 m 2 is equal to (1) 6 (2) 9 (3) 10 (4) 12
9. Let the eccentricity of Ellipse, 2 2 22 1,()xyab ab +=> is 1 4 . If the Ellipse passes through 2 4,3 5
. Then a 2+b 2 = _____.
(1) 29 (2) 31 (3) 32 (4) 34
10. If the points of intersection of Ellipse 2 2 2 1 16 xy b += and x 2 + y 2 = 4 b , b >4 lie on y2 = 3x2. Then b = ______. (1) 10 (2) 6 (3) 5 (4) 12
11. Let L be a tangent to y 2 = 4 x –20 at(6, 2). If L is also tangent to 2 2 1 2 xy b += . Then b = _____.
(1) 14 (2) 20 (3) 11 (4) 12
12. Two points A and B are ()() 7,0,7,0 and P is any point on, 9x2+16y2 = 144. Then PA+P B = (1) 16 (2) 8 (3) 6 (4) 9
13. If the tangents on Ellipse, 4x2+y2 = 8 at (1, 2) and (a, b)are perpendicular to each other, then a2 =
(1) 4 17 (2) 128 17 (3) 2 17 (4) 64 17
14. If tangent and normal at point P (in first quadrant) to ellipse 2 2 1 2516 xy+= intersect major axis at T and N respectively in such a way that ratio of area of ∆PTN and ∆PSS' is 91 , 60 then area of ∆PSS' is ( S and S' are foci)
(1) 63sq.units (2) 123sq.units
(3) 43sq.units (4) 33sq.units
15. A normal to the ellipse 2 2 22 1 xy ab += is tangent to circle x2 + y2 = r2 if a2 + b2 = 25 then possible value of r can be (1) 3 (2) 6 (3) 16 (4) 26
16. If the length and breadth of a rectangle of maximum area that can be inscribed in an ellipse 2 2 22 1 xy ab += are 82 and 42 respectively then the eccentricity of that ellipse
(1) 1 2 (2) 3 2
(3) 1 4 (4) 1 3
17. The set of values of m for which it is possible to draw the chord 1 ymx=+ to the curve
x2 + 2xy + (2 + sin2α)y2 = 1, which subtends a right angle at the origin for some value of α is
(1) [2, 3] (2) [0, 1] (3) [1, 3] (4) none of these
18. The radius of circle passing through foci of 2 2 1 169 xy+= , having centre at (0, 3)is
(1) 4 (2) 3 (3) 1 2 (4) 7 2
19. If the length of major axes is n times length of minor axes in an Ellipse. Then eccentricity is
(1) 1 n n (2) 2 1 n n (3) 2 2 1 n n (4) 2
20. The normal at one end of latus rectum of Ellipse 2 2 22 1 xy ab += passes through other end of minor axis then e 2 = (1) tan15° (2) tan18° (3) 2sin36° (4) 2sin18°
Section – B
21. If L be the length of common tangent to the ellipse 2 2 1 254 xy+= and the circle x2+y2 = 16 intercepted by the coordinate axis, then 3 2 L is _____.
22. Let ∆1 be the area of triangle PQR inscribed in an ellipse 2 2 22 1 xy ab += (a>b) and ∆ 2 be the area of triangle P'Q'R' whose vertices are the points lying on the auxiliary circle corresponding to the points P, Q, R respectively. If the eccentricity of the ellipse is 43 7 , then the ratio 2 1 ∆ ∆ ______.
23. Let S and S ' be two foci of the ellipse 2 2 22 1 xy ab += . Chords AB, CD are passing through the foci S and S' respectively. If A=( a cos a , b sin a ), B=( a cos b , b sin b ), C=(acosg, bsing), and D=(acosd, bsind), then the value of 9tantantantan 2222 αβγδ = _____.
24. If the tangent drawn at a point ( t2, 2t) on the parabola y2 = 4x is same as the normal drawn at a point () 5cos,2sinφφ on ellipse 4 x 2 +5 y 2 = 20 then the value of 5t2 = _____.
25. The area of the quadrilateral formed by the tangents at the end points of latusrectum to the ellipse 2 2 1 95 xy+= is _____.
CHAPTER TEST – JEE ADVANCED
2019 P1 Model
Section – A
[Multiple Correct Option MCQs]
1. Let PA and PB are two tangents drawn from point P, outside the ellipse 2 2 22 1 xy ab +=
(a > b). If H 1 and H 2 are reflections of focii F1 and F2 respectively in the tangents PA and PB, then which of the following option(s) is/are correct?
(1) 2121 PHFPFH∠=∠
(2) midpoint of line joining F1H1 lies on a circle of radius a
(3) ∆ PF2H1 and ∆ PF1H2 are similar (4) ∆ PF2H1 and ∆ PF1H2 are congruent
2. If the locus of P(x,y)satisfying the condition
()()()() −+−+++−= 2222 414112xyxy
represents a conic then
(1) Its eccentricity is 2 3
(2) Distance between its foci is 8
(3) Length of its minor axis is 12
(4) Length of its latusrectum is 20 3
3. If a tangent of slope 1 3 of the ellipse 2 2 22 1 xy ab += (a > b > 0) is normal to the circle x2+y2+2x+2 y+1=0 then which of the following is/are CORRECT?
(1) maximum value of ab is 2 3
(2) 2 ,2 5 a
(3)
(4) maximum value of ab is 1
4. If a tan ge nt of slope 1 3 of the ellipse += 2 2 22 1 xy ab (a > b > 0) is normal to the circle x2 + y2 + 2x + 2y +1 = 0, then which of the following is/are correct ?
(1) maximum value of ab is 2 3
(2)
2 ,2 5 a
(3)
2 ,2 3 a
(4) maximum value of ab is 1
5. Tangents are drawn from any point on the ellipse 2 2 1 94 xy+= to the circle x2+y2 = 1 and respective chord of contact always touces a conic C. Then, which of the following is/are true?
(1) minimum distance between C and the given ellipse is 3 2
(2) maximum distance between C and the given ellipse is 10 3
(3) eccentricity of C is 5 3
(4) () () = ecentricity of 1 eccentricity of given ellipse C
6. The line ()+−=< 2 311, 1 pxypp for different values of p, touches an ellipse whose
(1) eccentricity is 2 3
(2) length of latus rectum is 2 9
(3) eccentricity is 22 3
(4) length of latus rectum is 4 9
Section – B
[Numerical Value Questions]
7. Let R1 and R2 be the radii of the circles C 1 and C2 touching the ellipse 3 x2+4y2 = 7 at the point P(1, 1)and also touching x - axis; (R1>R2), Let the point Q(a, b)on the circle C2 which is farthest form the circle C1, then the value of 3a+9b is _______
8. Let C be the largest circle centred at (2, 0) and inscribed in the ellipse += 2 2 1 3616 xy . If (1, α) lies on C, then 10α2 is equal to
9. If the co-ordinates of two points A and B are respectively ( 7 ,0) and (− 7 ,0) P is any point on the conic, 9x2 + 16y2 = 144, then PA + PB is equal to_____
10. Let the line 2 y =– x + k is tangent to the ellipse () 2 2 2 2 111 11 xya a +=> which cuts its auxiliary circle at points A and B such that 90 AOB ∠= If e is eccentricity of ellipse, then the value of 8e2_____.(O being origin)
11. Locus of foot of perpendicular drawn from the centre (0, 0) to any tangent on the ellipse kx 2 +3 y 2 = 2, satisfies the equation 3 kx 2+4 y 2–6( x 2+ y 2) 2 = 0 then the value of 2 1 e (where e is eccentricity of the ellipse) is ______.
12. The number of tangent(s) to the circle x2+y2 = 3 which are normal to the ellipse 2 2 1 94 xy+= is/are _____.
13. Slope of normal drawn to the ellipse at a point P is 3 4 and eccentricity of the ellipse is 1 3 If this normal makes an acute angle b with its focal chord through P then the absolute value of 5sinβ is _______
14. Let Δ1 be the area of a triangle PQR inscribed in an ellipse and Δ 2 be the area of the triangle P'Q'R' whose vertices are the points
lying on the auxiliary circle corresponding to the points P, Q, R respectively. If the eccentricity of the ellipse is 43 7 then the ratio ∆ ∆ 2 1 is equal to
Section – C
[Single Correct Option MCQs]
15. AB is common tangent to the ellipse 2 2 22 1 xy ab += and the circle x 2 +y 2 = r 2 (b<r<a), The chord P Q of the circle is a focal chord to the ellipse. If PQ is parallel to AB, then its length is equal to (1) 22ab + (2) 22 1 ab + (3) 2a+2b (4) 2b
16. Point ‘O‘ is the centre of the ellipse with major axis AB and minor axis CD. Point F is one focus of the ellipse. If OF = 6 and the diameter of the inscribed circle of triangle OCF is 2 , then the product AB.CD is equal to ____
(1) 65 (2) 52 (3) 78 (4) 72
17. The locus of the midpoint of chords of the ellipse 2 2 22 1()xyab ab +=> passing through the point (2a, 0) (1) 2 2 22 () 1 xay ab += (2)
2 2 22 () 1 xay ab −= (3) 2 2 22 () 1 xay ab −=− (4)
2 2 22 () 1 xay ab +=−
18. If circumcentre of an equilateral triangle inscribed in 2 2 22 1 xy ab += with vertices having eccentric angles α, β, g respectively is (x1, y1) then CoscossinsinαβαβΣ⋅+∑ is (1) 22 11 22 99 3 2 22 xy ab +− (2)
22 11 22 5 2 22 xy ab +− (3) 22 11 22 5 9 99 xy ab +− (4)
22 11 22 1 10 xy ab +−
ANSWER KEY
JEE Main Level
Theory-based Questions
JEE Advanced Level
Brain Teasers
Flashback
Chapter Test – JEE Main
Chapter Test – JEE Advanced
HYPERBOLA CHAPTER 16
Chapter Outline
16.1 Introduction to Hyperbola
16.2 Different Forms of Hyperbola
16.3 Tangent and Normal to Hyperbola
16.4 Parametric Equations
16.5 Pair of Tangents
16.6 Asymptotes
16.7 Rectangular Hyperbola
Studying the equation of a hyperbola is essential for understanding its properties, such as asymptotes, vertices, foci, and eccentricity. This knowledge enables solving real-world problems in various fields like physics, engineering, and astronomy, where hyperbolas model trajectories, orbits, and wave propagation with precision and accuracy.
16.1 INTRODUCTION TO HYPERBOLA
The locus of a point whose distances from a fixed point and a fixed straight line are in constant ratio e, where e1 > , is called a hyperbola.
Here the fixed point is called focus, fixed line is called directrix, and e is called eccentricity.
Equation of hyperbola whose focus is S(x1,y1) directrix is lx+my+n = 0, eccentricity e is 222 SPePM =
A second-degree non-homogeneous equation in terms of x,y: 222220 axhxybygxfyc +++++= represents a hyperbola, if 2 0 hab−> and 222 20abcfghafbgch ∆=+−−−≠
Equation of hyperbola in the standard form:
The equation of a hyperbola in the standard form is 2 2 22 1 xy ab −=
If () , xy is a point on the curve, then ()() ,,, xyxy , and () , xy also lie on the curve. So, the curve is symmetric about both coordinate axes and symmetric about origin. Hence, the centre of the hyperbola is origin. If 0 y = , then xa =± . Therefore, the curve cuts the x-axis at two points () ,0Aa and () ,0Aa ′ .
The line segment on the line through the foci of the hyperbola with its end points on the hyperbola is called transverse axis . The line segment AA′ is called transverse axis and its length is 2a
The line segment through the centre of the hyperbola and perpendicular to the transverse axis with its end points () 0, Bb and () 0, Bb ′ is called conjugate axis. The line BB ′ is called conjugate axis and its length is 2b .
From the equation of hyperbola, 22 b yxa a =±− . It means that y is not real for all axa−≤≤ . It means, hyperbola does not pass through the origin, and does not meet the y-axis. It does not exist between the lines xa = and xa =− . The curve is not bounded on both the sides of the axes.
The focus of the hyperbola is () ,0 Sae and the corresponding directrix is a x e = . By the symmetry of the curve, there exists second focus () ,0 Sae ′ and corresponding directrix
a x e =− , Here
22ab e a + =
For the hyperbola, there exists two foci, two directrices, two latusrecta, one transverse axis and one conjugate axis, two vertices, one centre, etc. A chord passing through either of the foci is called focal chord
Important points
1. The locus of a point, the difference of whose distances from two fixed points is a constant k , provided the distance between the fixed points is more than k is a hyperbola.
2. The difference of focal distances of any point on the hyperbola is constant. Let P be any point on the hyperbola. Then
2 SPSPa ′ −= , where 2a is the length of the transverse axis and and SS ′ are foci.
3. The angle made by the chord joining the end points of transverse axis and conjugate axis of hyperbola with x-axis
is 1 22 sin b ab
+
4. A chord passing through any point P on the hyperbola and perpendicular to transverse axis is called double ordinate of the point P .
5. A focal chord of a hyperbola perpendicular to the transverse axis of the hyperbola is called latus rectum . There will be two latus recta in hyperbola.
6. If PSQ is a focal chord of the hyperbola, then 112 SPSQSL += , where SL is semi latus rectum.
In a hyperbola, if the length of the transverse axis () 2a is equal to the length of the conjugate axis () 2b then the hyperbola is called rectangular hyperbola
The equation of rectangular hyperbola is 222 xya −=
The eccentricity of rectangular hyperbola is 22 2 aa e a + ==
The hyperbola whose transverse axis and conjugate axis are, respectively, conjugate axis and traverse axis of given hyperbola, is called the conjugate hyperbola of the given hyperbola.
The equation of hyperbola conjugate to 2 2 22 1 xy ab −= is 2 2 22 10. xy ab −+=
1. Find the equation of the hyperbola whose vertices are ()() 2,0,2,0 and the foci are ()() 3,0,3,0
Sol. Given: Vertices are ()() 2,0,2,0 and foci are ()() 3,0,3,0
Distance between the vertices is 24 a = and distance between the foci is 26 ae =
Hence, the eccentricity is 23 22 ae e a ==
The centre of the hyperbola is () 0,0 , and the transverse axis is along x-axis.
We know that ()222 1 bae=−
Hence, 2 9 415 4 b =−=
Therefore, the equation of the hyperbola is 2 2 1 45 xy −=
Try yourself:
1. If the equation () 22 244kxy−+= represents the hyperbola, then find the value of k
Ans: 2 k <
TEST YOURSELF
1. Equation of the hyperbola of eccentricity 3, and the distance between whose foci is 24, is
2. The eccentricity of x y 2 2 916 1 is (1) 17 16 (2) 5 4 (3) 5 3 (4) 7 4
3. If c is a real number and x c y c 2 2 12 7 1 represents a hyperbola, then (1) 7 < c < 12 (2) c < 7 (3) c > 12 (4) c < 7 or c > 12
4. If the equation (10 x – 5) 2 + (10 y – 4) 2 = l2(3x + 4y –1)2 represents a hyperbola, then (1) –2 < l<2 (2) l>2 (3) l< –2, l>2 (4) 0 < l<2
5. The equation x K y K 2 2 75 1 represents a hyperbola if (1) 5 < K < 7 (2) K < 5 or K > 7 (3) K > 5 (4) K ≠ 5, K ≠ 7
6. S–I: The number of integral values of K for which the equation 2 2 1 3210 xy KK += represents a hyperbola is 10.
S–II: Th e above equation represents a rectangular hyperbola for K = 3. Which of above statements is true
(1) Only I (2) Only II
(3) Both I and II (4) Neither I nor II
Answer Key (1) 4 (2) 3 (3) 4 (4) 3 (5) 1 (6) 2
16.2 DIFFERENT FORMS OF HYPERBOLA
In this section, we will discuss different forms of hyperbola, depending on the position of the transverse axis.
16.2.1 Equation of Hyperbola Whose Axes are along Coordinate Axes
1. The equation of hyperbola whose transverse axis is along the x-axis is 2 2 22 1 xy ab −= . Y
L
2. For the hyperbola 2 2 22 1 xy ab −=
i. Centre is () 0,0 C
ii. The transverse axis is along x-axis and equation of transverse axis is 0 y = .
iii. The conjugate axis is along y-axis and equation of conjugate axis is 0 x = .
iv. The length of transverse axis is 2a and the length of the conjugate axis is 2b .
v. The ends of the transverse axis are vertices ()() ,0and,0AaAa ′
vi. Eccentricity is 22ab e a + = .
vii. Foci are ()() ,0,,0SaeSae ′ and the distance between the foci is 2ae .
viii. If P is any point on the Hyperbola, then 2 SPSPa ′ −=
ix. The length of the latusrectum is 2 2b a and the equations of the latusrecta are x = ae, x = –ae
x. Ends of the latusrectum are 2 , b ae a ±±
xi. Equations of directrices are a x e =± and distance between the directrices is 2a e
3. The equation of hyperbola having transverse axis along the y–axis and centre at origin is 2 2 22 1 xy ab −=−
4. For the Hyperbola 2 2 22 1 xy ab −=− :
i. Centre is () 0,0 C
ii. The transverse axis is along y-axis and equation of transverse axis is 0 x =
iii. The conjugate axis is along x-axis and equation of conjugate axis is 0 y = Y L’ L S X B Z’ B’ S’ A’ A C Z
iv. The length of transverse axis is 2b and the length of the conjugate axis is 2a
v. The ends of the transverse axis are vertices ()() 0,,0, BbBb ′
vi. Eccentricity is 22ba e b + =
vii. Foci are ()() 0,,0, SbeSbe ′ and distance between the foci is 2 SSbe ′ =
viii. If P is any point on the hyperbola, then 2 SPSPb ′ −=
ix. The length of the latus rectum is 2 2a b and the equations of the latus recta are , ybeybe ==−
x. Ends of the latus rectum are 2 , abe b
xi. Equations of directrices are b y e =±
The hyperbola, whose transverse and conjugate axes are, respectively, the conjugate and transverse axes of a given hyperbola, is called the conjugate hyperbola of the given
hyperbola. The hyperb olas 2 2 22 1 xy ab −= and 2 2 22 1 xy ab −=− are conjugate hyperbolas to each other. If 12 , ee are eccentricities of two conjugate hyperbolas, then 2222 1212 eeee +=
2. Find the equation of the hyperbola whose vertices are ()() 5,0,5,0 and whose one of the foci lies on the line 3525 xy−=
Sol. The vertices ()() 5,0,5,0 lie on the x-axis, so that the hyperbola is in standard form, and the equation of the hyperbola can be taken as 2 2 22 1 xy ab −=
Here 5 a = . The foci () ,0 ae ± lies on the line 3525 xy−= So, ()() 5 350253525 3 aeee −=⇒=⇒=
Hence, 22 2 2 2 2 2525 925 400 9 ab e a b b + = + = = Therefore, the equation of the required hyperbola is 2 2 9 1 25400 xy −=
Try yourself:
2. Find the equation of the hyperbola in the standard form, which is passing through the point () 2,1 and has eccentricity e2 = . Ans: 22 311 xy−=
16.2.2 Equation of Hyperbola Having Axes
Parallel to Coordinate Axes
1. The hyperbola having transverse axis parallel to the x-axis and centre at () , hk is ()() 22 22 1 xhyk ab −= , x = h y = k
2. For the Hyperbola ()() 22 22 1, xhyk ab −=
i. Centre is () , Chk
ii. The transverse axis is parallel to x-axis and equation of transverse axis is y – k = 0.
iii. The conjugate axis is parallel to y-axis and equation of conjugate axis is x – h = 0.
iv. The length of transverse axis is 2a and the length of the conjugate axis is 2 b.
v. The ends of the transverse axes are vertices ()() ,,, AahkAahk ′ +−+ .
vi. The ends of the conjugate axes are ()() ,,, BhbkBhbk ′ +−+
vii. Eccentricity is 22ab e a + = .
viii. Foci are ()() ,,, ShaekShaek ′ +−
ix. The length of the latus rectum is 2 2b a and the equations of the latus recta are , xhaexhae −=−=−
x. Equations of directrices are xha e −=±
3. The hyperbola having transverse axis
parallel to the y-axis and centre at () , hk is ()() 22 22 1 xhyk ab −=− x = h
4. For the hyperbola ()() 22 22 1 xhyk ab −=−
i. Centre is C(h, k).
ii. The transverse axis is parallel to y-axis and equation of transverse axis is 0 xh−= .
iv. The length of transverse axis is 2b and the length of the conjugate axis is 2a .
v. The ends of the transverse axis are vertices ()() ,,, BhbkBhbk ′ +−+ .
vi. The ends of the conjugate axis are ()() ,,, AahkAahk ′ +−+
vii. Eccentricity is 22ba e b + = .
viii. Foci are ()() ,,, ShkbeShkbe ′ +− .
ix. The length of the latus rectum is 2 2a b and the equations of the latus recta are , ykbeykbe −=−=− .
x. Equations of directrices are ykb e −=±
3. Find the equation of the hyperbola whose vertices are () 4,1 and () 6,1 and one of the focal chords is 220.xy−−=
Sol. The vertices () 4,1 and () 6,1 lie on the line 10 y −= , and the distance between the vertices is 2105 aa=⇒= .
Centre of the hyperbola is () 1,1 , which is midpoint of vertices.
The foci are () 1,1 ae ±
The point () 1,1 ae + lies on 260xy−−= ,
so that 7 12607 5 aeaee +−−=⇒=⇒=
Hence, ()222 49 125124 25 bae =−=−=
Therefore, the equation of the hyperbola is ()() 22 11 1 2524 xy −=
iii. The conjugate axis is parallel to x-axis and equation of conjugate axis is 0 yk−=
Try yourself:
3. Find the equation of the hyperbola having vertices ()() 2,2,2,4 , and e3 =
Ans: )()( 22 21 1 729 xy −=−
16.2.3 Equation of Hyperbola Referred to Two Perpendicular Lines
Let the conjugate and transverse axes of the hyperbola be along two mutually perpendicular lines, 1111 0 Laxbyc≡++= and 2112 0 Lbxayc≡−+= , respectively. Let a and b be the lengths of semi-transverse axis and semi-conjugate axes, respectively.
The equation of the hyperbola is 22 111112 22 22 11 11 22 1 axbycbxayc abab ab ++−+ ++ −=
For the above hyperbola:
1. The centre of the hyperbola is the point of intersection of two lines 1 0 L = and 2 0. L =
i. The transverse axis is along the line 2 0 L = and the conjugate axis is along the line 1 0 L =
ii. The length of the transverse axis is 2a and the length of the conjugate axis is 2b.
iii. Foci are the points of intersection of 22 111 0 Laeab±+= and 2 0 L = .
iv. Equations of directrices are 22 1111 0 a axbyab e +±+=
4. Find the transverse axis, conjugate axis, centre, and eccentricity of the hyperbola ()() 22 421922180 xyxy −+−++=
Sol. Equation of the Hyperbola is ()() 22 2122 1 4520 xyxy −+++ −=
The transverse axis is 220 xy++= and the conjugate axis is 210xy−+=
Centre is the point of intersection of the above two lines, so it is () 1,0 . The eccentricity is 22 9413 3 9 ab e a ++ ===
Try yourself:
4. Find the equation of the hyperbola whose lengths of transverse and conjugate axes are 6 and 26 respectively, their equations are 330xy−+= and 310 xy+−= respectively.
Ans: )()( 22 3133 1 9060 xyxy +−−+ −=
Notation:
Here after the following notation will be adopted throughout this chapter.
1. 2 2 22 1 Sxy ab =−−
2. 11 1 22 1 Sxxyy ab =−−
3. 22 11 11 22 1 Sxy ab =−−
4. 1212 12 22 1 Sxxyy ab =−−
16.2.4 Position of a Point With Respect to Hyperbola
A hyperbola divides the plane into three regions. The region containing the focus, called the interior of the hyperbola. The
Interior region
16: Hyperbola
region consisting of the curve itself. The region containing the remaining portion of the plane, called the exterior of the hyperbola. C L X Y Interior region
Exterior region
Exterior region
P(x1,y1)
Q(x1,y2)
Let () 11 , Pxy be any point in the plane of hyperbola S = 0,then
i. () 11 , Pxy lies outside the hyperbola
⇔ 11 0 S < .
ii. () 11 , Pxy lies on the hyperbola ⇔
11 0 S =
iii. () 11 , Pxy lies inside the hyperbola ⇔
11 0 S >
TEST YOURSELF
1. The locus of the point , 22
is a hyperbola with eccentricity
(1) 3 (2) 3 (3) 2 (4) 2
2. The equation of the hyperbola whose centre is (1, 2), and one focus is (6, 2) and transverse axis 6 is
(1) 16(x –1)2 – 9(y – 2)2 = 144
(2) 9(x –1)2 – 16(y – 2)2 = 144
(3) 16(x –1)2 – 25(y – 2)2 = 200
(4) 25(x –1)2 – 16(y – 2)2 = 200
3. The equation of the hyperbola with its transverse axis is parallel to x –axis, and its centre is (–2, 1) the length of transverse axis is 10 and eccentricity 6/5 is
(1) ()() 22 32 1 169 xy−+ −=
(2) ()() 22 21 1 2511 xy+− −=
(3) ()() 22 32 1 69 xy −=
(4) ()() 22 23 1 1619 xy −=
4. The vertices of a hyperbola are (2, 0), (–2, 0) and the foci are (3, 0), (–3, 0). The equation of the hyperbola is (1)
5. If m is a variable, the locus of the point of intersection of the li nes 32 xym −= and 1 32 xy m += is
(1) a parabola (2) an ellipse (3) a hyperbola (4) Straight line
6. If e and e ’ are the eccentricities of the ellipse 5x2 + 9y2 = 45 and the hyperbola 5x2 – 4y2 = 45, then e e’ = (1) 9 (2) 5 (3) 4 (4) 1
7. The vertices of the hyperbola x2 – 3y2 + 2x + 12y + 1 = 0 are
Chords on a hyperbola are line segments connecting any two points on the curve. When it is extended as line if one point approaches the other along the curve. Then the chord becomes a tangent to that point on the curve. Normals are perpendicular lines to tangents at their point of contact. The equation of the chord joining two points () 11 , Pxy and () 22 , Qxy on the Hyperbola 0 S = is 1212SSS +=
16.3.1 Tangent and Normal to the Hyperbola
Position of a line with respect to the hyperbola: consider the hyperbola 2 2 22 1 xy ab −= and the line ymxc =+
i. intersect the hyperbola, if 2222 camb >−
ii. touches the hyperbola, if 2222 camb =−
iii. does not intersect the hyperbola, if 2222 camb <−
The equation of tangent to the hyperbola 0 S = at () 11 , Pxy is 1 0. S = The slope form of tangent to the hyperbola 2 2 22 1 xy ab −= is 222 ymxamb =±− and its point of contact is 22 , amb cc where 222 camb =±−
The condition for the line 0 lxmyn++=
to be tangent to the Hyperbola 2 2 22 1 xy ab −= is 22222 albmn −= and the point of contact is 22 , albm nn
Important points:
The feet of the perpendicular drawn from either of the foci to any tangent to the hyperbola lie on a circle concentric with the hyperbola. This circle is called the auxiliary circle of the hyperbola.
1. The auxiliary circle of hyperbola is a circle with transverse axis as diameter
2. The equation of auxiliary circle of the hyperbola 2 2 22 1 xy ab −= is 222 xya += . The equation of auxiliary circle of 2 2 22 1 xy ab −=− is 222 xyb +=
3. If tangent at a point P to the hyperbola intersect the transverse axis at T . If N is foot of the perpendicular from P on transverse axis, then the circle having NT as diameter intersect the auxiliary circle orthogonally.
4. The tangents at the extremities of the latursectum of a hyperbola intersect on corresponding directrix.
5. The tangents at the extremities of the focal chord of a hyperbola intersect on corresponding directrix.
6. The portion of the tangent to a hyperbola between the point of contact and the directrix subtends right angle at corresponding focus.
7. The product of the perpendiculars from the foci on any tangent to a hyperbola is equal to the square of the semi conjugate axis and the feet of these perpendiculars lies on the auxiliary circle.
8. If any tangent to a hyperbola intersect the tangents at the vertices at T and T ′ , then the circle considering TT ′ as diameter passes through the foci.
9. The circle on any focal distance of Hyperbola as diameter touches the auxiliary circle.
10. A circle cuts a Hyperbola in four real or imaginary points, then the sum of the eccentric angles of these four concyclic points on the Hyperbola is an even multiple of π .
5. Find the equation of tangent to the Hyperbola 22 236 xy−= which makes an angle 60° with transverse axis.
Sol. Equation of the Hyperbola is 2 2 1 32 xy −=
The slope of the tangent is tan603 m =°=
The equation of tangent in the slope form is () 222 3332 37 ymxamb x yx =±− =±− =±
Therefore, the equation of the tangent is 37yx=±
Try yourself:
5. If 34122 xy+= is a tangent to the hyperbola 2 2 2 1 9 xy a −= for some aR ∈ then find the distance between the foci of hyperbola.
Ans: 257
The equation of normal to the hyperbola 2 2 22 1 xy ab −= at () 11 , Pxy is 2 2 22 11 axbyab xy +=+
Equation of normal drawn at 2 , b ae a
is 3 xeyae −=
The condition for the line 0 lxmyn++=
to be a normal to the hyperbola 2 2 22 1 xy ab −= is ()2 22 22 222 abab lmn + −=
The slope form of normal to the hyperbola 2 2 22 1 xy ab −= is () 22 222 mab ymx abm + =±
Common Tangent and Common Normal:
The equation of common tangent to the hyperbola 2 2 22 1 xy ab −= and 2 2 22 1 xy ab −=− are 22yxab =±±−
The length of the common tangent is 22 22 2 ab ab +
16.3.2 Normal Line to Hyperbola
Let 0 S = be a hyperbola and P be a point on the hyperbola 0 S = . The line passing through P and perpendicular to the tangent of 0 S = at P is called the normal to the hyperbola
0 S = at P
At most, four normals can be drawn from a point to a hyperbola. 90-θ θ θθ 90-θ P(a sec θ, b tan θ) x = a –e (ae,0) (–ae,0) F2 F1 x = a e
Let S and S ′ be two foci of hyperbola. The tangent at P to the hyperbola is internal
angular bisector of SPS ∠′ and normal at P to the hyperbola is external angular bisector of SPS ∠′ .
Reflection property of a hyperbola:
If an incoming light ray aimed towards one focus and strike convex side of the hyperbola, then it will get reflected toward the other focus S ′
Try yourself:
6. Find a point on the hyperbola 22339xy−= when the normal is parallel to the line 652 xy−=
Ans: )( 53,23
TEST YOURSELF
1. If the line x + y + k = 0 is a normal to the hyperbola 2 2 1 94 xy −= , then k = (1) 5 13 ± (2) 13 5 ± (3) 13 5 ± (4) 5 13 ±
Confocal Curves:
If an ellipse and a hyperbola have same foci, they cut at right angles at any of their point of intersection 2 2 22 1, xy ab += 2 2 22 1, xy ab λλ −= (a >l> b) are confocal and orthogonal to each other.
6. If the normal at one end of a latus rectum of the Hyperbol a 2 2 2 1 32 xy b −= passes through one end of the conjugate axis, then 4 2 e 1e
Sol. Equation of the normal is
Given that, it is passing through the point () 0, b Hence, 22abab =+ , substitute () 2 1 bae=− and then simplify. It implies that 4 2 1 1 e e =
2. If the line lx + my = 1 is a normal to the hyperbola 2 2 22 1 xy ab −= , then 22 22 ab lm
3. 3x + 4y – 7 = 0 is normal to 4x2 – 3y2 = 1 at the point
(1) (–3, 4) (2) (1, 1)
(3) (2, 1 4 ) (4) (5, –2)
4. The equations of the tangents to the hyperbola 2 x 2 – 3 y 2 = 6 which are perpendicular to the line x – 2y + 5 = 0 are
(1) – 211 0 xy ±= (2) 2 10 0 xy+±=
(3) 521 0 xy+±= (4) 631 0 xy+±=
5. The condition that the line x = my + c may be a tangent of 2 2 22 1 xy ab −=− is
(1) c2 = a2m2 – b2
(2) c2 = a2 – b2m2
(3) c2 = b2 – a2m2
(4) c2 = b2m2 – a2
6. Equations of tangents to the hyperbola 4 x 2 – 3 y 2 = 24 which makes an angle 30° with y–axis are
(1) 310 xy−=± (2) 310 xy−=±
(3) 35 xy−=± (4) 35 xy−=±
Answer Key
(1) 2 (2) 3 (3) 2 (4) 2 (5) 4 (6) 1
16.4 PARAMETRIC EQUATIONS
Let P be a point on the Hyperbola with centre at C and NP is the ordinate of the point P . The tangent from N to the auxilary circle meets the auxiliary circle in P ′ . If measure of angle NCP ′ is θ then θ is called the eccentric angle of P . The point P ′ is called corresponding point of P .
() sec,tanPabθθ is () cos,sinPaa′θθ
The equation of the chord joining two points , αβ on the hyperbola 2 2 22 1 xy ab −= is cossincos 222 xy ab
The condition for the chord joining , αβ to be focal chord is
i. cosecos 22
ii. tantan(or)1e1e 221e1e αβ−+
If the chord joining the points α and β on the hyperbola 0 S = cuts the major axis at a distance d units from the center, then tantan 22 da da αβ−
Eccentric angles of the extremities of latusr ectum of an hyperbo la 2 2 22 1 xy ab −= are 1 tan b ae
Le t θ be the eccentric angle of a point () , Pxy on the hyperbola 2 2 22 1, xy ab −= then sec,tan,xayb=θ=θ where 02≤θ<π .
The parametric equations of the hyperbola 2 2 22 1 xy ab −= a re sec,tanxayb=θ=θ ; 02≤θ<π
The point () sec,tanPabθθ is a point on the hyperbola 2 2 22 1 xy ab −= , whose eccentric angle is θ . The corresponding point of
The equation of tangent at () P θ to the hyperbola 2 2 22 1 xy ab −= is sectan1xy ab θ−θ=
The equation of the normal to the hyperbola 2 2 22 1 xy ab −= at () P θ is 22 sectan axbyab +=+ θθ 0,, 2 π θ≠π
The focal distance of a point ()() 11,sec,tanPxyab=θθ is 1 sec SPexaaea =−=θ− and 1 sec SPaexaae ′=+=+θ
Co-normal points:
The points on the hyperbola, at which the normals drawn to the hyperbola passes through a given point are called co normal points.
At most four normals can be drawn from any point to the hyperbola and the sum of the eccentric angles of their feet is an odd multiple of π . If 1234 ,,, θθθθ are feet of normals to the hyperbola then () 1234 21 n θ+θ+θ+θ=+π .
If 1234 ,,, θθθθ are eccentric angles of four points on the hyperbola 2 2 22 1 xy ab −= . The normals at which are concurrent, then () 12 cos0∑θ+θ= and () 12 sin0∑θ+θ= .
7. If , αβ are the eccentric angles of the extremities of a focal chord of the hyperbola 2
−= , then what is the value of
Sol. We know that cosecos
and 22 5 2 ab e a + == Hence, 5cos 2
is 2cos 2
Try yourself:
7. Find the equation of the normal at 3 θ=π to the hyperbola 22 3412 xy−= . Ans: 7 xy+=
TEST YOURSELF
1. Equation of normal to 9x2 –25y2 = 225 at θ = 4 π is
(1) 5 3 2 34 2 xy+= (2) 5 2 34 2 xy+= (3) 5 3 34 2 xy+= (4) 5 3 2 34 2 xy−= 2. Locus of feet of perpendicular from (5, 0) to the tangents of 2 2 1 169 xy −= is
3. Equation of the tangent to the hyperbola 4x2 – 9y2 = 1 with eccentric angle 6 π is (1) 4 3 3 xy+= (2) 4 3 3 xy−= (3) 3 4 3 xy−= (4) 3 4 5 xy−=
4. If α and b are two points on the hyperbola 2 2 22 1 xy ab −= and the chord joining these two points passes through the focus (ae, 0), then cos 2 e αβ = (1) cos 2 αβ + (2) cos 2 αβ (3) 22 cos 3 αβ + (4) cos 4 αβ +
5. If the normal at θ on the hyperbola 2 2 22 –1xy ab = meets the transverse axis at G, then AG A’G = (1) a2(e4 sec2θ – 1) (2) a2(e4 sec2θ + 1) (3) b2(e4 sec2θ – 1) (4) none
Answer Key
(1) 1 (2) 2 (3) 2 (4) 1 (5) 1
16.5 PAIR OF TANGENTS
The combined equation of a pair of tangents drawn from a point () 11 , Pxy to the hyperbola 0 S = is 2 111SSS =⋅
If m 1 and m 2 are the slopes of the two tangents drawn from a point () 11 , Pxy to the hyperbola, then 12 , mm are the roots of the equation ()() 222 22 1 111 20xamxymyb−−++=
If m 1 and m 2 are the slopes of the two tangents drawn from a point () 11 , Pxy to the hyperbola, then 11 12 22 1 2 xy mm xa += and 22 1 12 22 1 yb mm xa + =
The locus of the point of intersection of two tangents to the hyperbola 2 2 22 1 xy ab −= which makes an angle α with one another is ()() 2 22222222222 4 cot xyabbxayab +−+=−++α
If θ is the angle between two tangents, then 11 2222 11 2 tan abS xyab θ= +−+
1. The locus of the point of intersection of perpendicular tangents to a hyperbola is a circle, this circle is called director circle.
2. The equation of director circle to the hyperbola 2 2 22 1 xy ab −= is 2222 xyab +=−
3. The equation of director circle to the hyperbola ()() 22 22 1 xhyk ab −= is ()() 22 22xhykab −+−=−
4. The tangents at the points P( α ) and Q( b ) intersect at the point cossin 22 , coscos 22 ab
5. Product of the perpendiculars from foci on any tangent to the hyperbola 2 2 22 1 xy ab −= is 2 b .
Chord of contact:
The line joining the points of contact of the tangents to a hyperbola 0 S = drawn from an external point P is called the chord of contact of P with respect to the hyperbola 0 S = .
The equation of chord of contact of () 11 , Pxy with respect to the hyperbola 0 S = is 1 0 S = .
Midpoint chord
The equation of a chord of hyperbola 0 S = whose midpoint () 11 , Pxy is 111SS = The midpoint of the chord 0 lxmyn++= of the hyperbola 2 2 22 1 xy ab −= is
8. Find the slopes of the tangents drawn from () 4,1 to the hyperbola 2226xy−= .
Sol. If m1 and m2 are the slopes of tangents drawn from () 11 , xy to the hyperbola 2 2 22 1 xy ab −= then () () 11 12 22 1 241 2 4 5 46 xy
Therefore, the slopes are 44 , 55
Try yourself:
8. If the chords of contact of () 11 , Pxy and () 22 , Qxy with respect to the hyperbola
2 2 22 1 xy ab −= are at right angles, then find the value of 12 12 xx yy Ans: 4 4 a b
TEST YOURSELF
1. Product of perpendiculars from the foci of
2 2 1 49 xy −= to 2 49ymxm=+− , where 3 2 m > is (1) 4 (2) 36 13 (3) 3 (4) 9
2. If m1 and m2 are slopes of the tangents to the hyperbola 2 2 1, 2516 xy −= which pass through the point (6, 2), then (1) 12 24 11 mm+= (2) 12 48 11 mm+= (3) 12 28 11 mm = (4) 12 11 20 mm =
3. Number of points from which perpendicular tangents can be drawn to the curve
5. The midpoint of the chord 4 x – 3y = 5 of the hyperbola 2x2 – 3y2 = 12 is (1) (2, 1) (2) (5, 5) (3) 1 1, 3
(4) 1 ,1 2
6. Locus of P such that the chord of contact of P with respect to y 2 = 4 ax touches the hyperbola x2 – y2 = a2 is
(1) x2 + 4y2 = 4a2 (2) 4x2 +
= 4
2 (3) x2 + 2y2 = 2a2 (4) 2x
7. The locus of the point of intersection of two tangents to the hyperbola 2 2 22 1 xy ab −= , which make an angle of 60° with one another, is
(1) (x2 + y2 – a
(2) (
3(
a2b2)
(4) x2 + y2 = a2 – b2
Answer Key
16.6 ASYMPTOTES
The tangents of the hyperbola which touch the hyperbola at the points at infinity are called asymptotes of the hyperbola.
The equations of the asymptotes to the hyperbola 2 2 22 1 xy ab −= are 0,0yyxx abab −=+= and the combined equation of these two asymptotes is 2 2 22 0 xy ab −=
When ba = , i.e., the asymptotes of rectangular hyperbola 222 xya −= are , yx =± which are at right angles.
S’ S X Y C A’ A
The e quations to the pair of asymptotes and the hyperbola differ by a constant.
The asymptotes of a hyperbola passes through the centre of the hyperbola.
Asymptotes are equally inclined to the axis of hyperbola.
Any straight line parallel to the asymptotes intersect the hyperbola at only one point.
The angle between the two asymptotes is
1 2tan b a
or2 1 sece .
The product of the perpendiculars from any point on the hyperbola to its asymptotes is 22 22 ab ab +
Asymptotes of a hyperbola and its conjugate hyperbola are same.
If H , C, and A are the equations of a hyperbola, its conjugate hyperbola and its pair of asymptotes respectively, then H + C = 2A.
The bisectors of the angles between the asymptotes are the transverse and conjugate axes.
9. Find the asymptotes of the hyperbola 22 4936. xy−=
Sol. The combined equation of asymptotes is 22 490 xy−=
Hence, the equations of asymptotes are 230,230 xyxy −=+=
Try yourself:
9. The asymptotes of a hyperbola having centre at () 1,2 are parallel to 230,320 xyxy +=+=
The hyperbola passes through the point () 5,3 . Find the equation of the hyperbola.
Ans: )( )( 238327154 xyxy +−+−=
TEST YOURSELF
1. The equation of the pair of asymptotes of the hyperbola 2x2 – y2 = 1 is
(1) 2x2 + y2 = 0
(2) 2x2 – y2 = 0
(3) x2 + 2y2 = 0
(4) x2 – 2y2 = 0
2. The angle between the asymptotes of the hyperbola xy = a2 is
(1) 30° (2) 60° (3) 45° (4) 90°
3. The eccentricity of the hyperbola with asymptotes 3x + 4y = 2 and 4x – 3y = 2 is
(1) 3 (2) 2 (3) 2 (4) 4
4. If the angle between the asymptotes of a hyperbola is 30°, Then e =
(1) 6 (2) 2
(3) 62 (4) 63
5. The asymptotes of the hyperbola 6x2 + 13xy + 6y2 – 7x – 8y – 26 = 0 are
(1) 2x + 3y – 1 = 0, 3x + 2y +2 = 0
(2) 2x + 3y = 1, 3x + 2y = 2
(3) 2x + 3y = 0, 3x + 2y = 0
(4) 2x + 3y = 3, 3x + 2y = 4
6. The asymptotes of a hyperbola are parallel to 3x + 2y = 0, 2x + 3y = 0, whose centre is at (1,2) and it passes through the point (5,3) . Then, its equation is
(1) 6x2 + 13xy + 6y2 – 37x – 38y – 56 = 0
(2) 6x2 + 13xy + 6y2 – 38x – 37y – 98 = 0
(3) 6x2 + 13xy + 6y2 – 38x + 37y – 98 = 0
(4) None
7. The product of the perpendicular from any point on the hyperbola 2 2 22 1 xy ab −= to its asymptotes, is
(1) 22 22 ab ab + (2) 22 22 ab ab
(3) 22 22 ab ab + (4) 22 22 ab ab
8. The product of the lengths of the perpendiculars from any point of the hyperbola x2 – y2 = 8 to its asymptotes is (1) 2 (2) 3 (3) 4 (4) 8
9. The asymptotes of the hyperbola xy = hx + ky are
(1) x = k, y = h (2) x = h, y = k
(3) x = h, y = h (4) x = k, y = k
10. The product of lengths of perpendiculars from any point on the hyperbola x2 – y2 = 16 to its asymptotes is
(1) 2 (2) 4 (3) 8 (4) 16
Answer Key
(1) 2 (2) 4 (3) 3 (4) 3
(5) 2 (6) 2 (7) 3 (8) 3
(9) 1 (10) 3
16.7 RECTANGULAR HYPERBOLA
A hyperbola in which ab = is called rectangular hyperbola, or a hyperbola whose asymptotes include a right angle.
The eccentricity of rectangular hyperbola is e2 =
Rectangular hyperbola of the form 222 xya −= : Focus Focus Vertex Vertex
Co-vertex Co-vertex
Conjugate axis Transverse axis Center Asymptote Asymptote
For the rectangular hyperbola 222 xya −= :
1. Asymptotes are 0,0xyxy+=−=
2. Centre is at origin.
3. Foci are () 2,0 a ±
4. Vertices are () ,0a ±
5. Equation of directrices are 2 a x =± .
6. Length of the latus rectum is 2a .
7. Parametric form is sec,tanxaya=θ=θ.
8. Equation of tangent is sectan xya θ−θ=
9. Equation of normal is 2 sectan xya += θθ
For the rectangular hyperbola of the form 2 xyc = :
For the rectangular hyperbola 2 xyc =
1. Equations of asymptotes are x0 = and y0 = .
2. Eccentricity is e2=.
3. Centre is at origin.
4. Foci are ()() 2,2,2,2 cccc .
5. The length of the latus rectum is 22c .
6. Directrices are 2 xyc +=± .
7. Parametric form is , c xctyt ==
8. The hyperbola conjugate to 2 xyc = is 2 xyc =−
The equation of the chord joining 1t and is ()1212 0 xyttctt+−+=
Equation of tangent at , c ct t
to the hyperbola is 2 x tytc +=
The tangent at 1t and 1t to the rectangular hyperbola intersect at 12 1212 2 2 , cttc tttt ++
Point form:
The equation of tangent at 11 (,) xy to the hyperbola 2 xyc = is 2 11 2 xyyxc += or 11 2 +=xy xy
The portion of the tangent drawn to 2 xyc = between the axes is bisected at the point of contact.
Area of triangle formed by tangent to 2 xyc = with coordinate axes is always a constant is equal to 2c2.
The equation of chord of contact of tangents drawn from a point () 11 , xy to the rectangular hyperbola is 2 11 2 xyyxc += .
The equation of normal to at () 11 , xy to the rectangular hyperbola is 22 1111 xxyyxy −=−
The equation of normal at , c ct t to the rectangular hyperbola 2 xyc = is 34 0 xtytctc−−+=
The equation of the normal at , c ct t
is a fourth degree equation in t . So, in general, maximum four normals can be drawn from a point to the hyperbola 2 xyc = . Point of intersection of normals at 1t and 2 t is
If the normal at the point 1t to the rectangular hyperbola 2 xyc = meets it again at the point t2, then 3 12 1 tt =− .
The equation of the chord of the hyperbola 2 xyc = , whose midpoint is () 11 , xy is 1111 2 xyyxxy +=
If a triangle is inscribed in a rectangular hyperbola, then its orthocentre lies on the hyperbola.
1. If 123 123 ,,,,, ccc PctQctRct ttt
are points on 2 xyc = , then the orthocentre of PQR∆ is 123 123 , c cttt ttt
2. 1234 1234 ,,,,,,, cccc PctQctRctSct tttt
are concyclic. Then the ortho centre of 4 4 , c PQRctS t
3. If a rectangular hyperbola xy = c 2 and a circle x2 + y2 + 2gx + 2fy + c = 0 intersect at four points ( x i , y i ), i = 1,2,3,4 the ,, 4422 ii xygf
∑∑ .
The mean position of the four concyclic points bisects the line joining centres.
The equation of the hyperbola having x =α and y =β as asymptotes can be taken as () () 2 xxk −α−β=
The angle between the asymptotes of a rectangular hyperbola is 90° .
The equation 222 xya −= is transformed to 2 xyc = when the axes are rotated through an angle of 45° , without changing the position of the point. Here, 2 2 2 a c = 1. 2222 and xyaxyc −== cut each other orthogonally.
2. If a circle and a rectangular hyperbola 2 xyc = meet at four points (),,,1,2,3,4, iii i c xycti t == , then 1234 1 tttt = 44 12341234 , xxxxcyyyyc ⇒==
TEST YOURSELF
1. The locus of middle points of normal chords of the rectangular hyperbola x2 – y2 = a2 is
(1) (x2 + y2)3 + 4a2x2y2 = 0
(2) (x2 – y2)3 + 4a2x2y2 = 0
(3) (x2 + y2)3 – 4a2x2y2 = 0
(4) (x2 – y2)2 – 4a2x2y2 = 0
2. If the circle x 2 + y 2 = r 2 intersects the hyperbola xy = c2 in four points (xi, yi ) for i = 1, 2, 3, and 4, then y1 + y2 + y3 + y4 = ___.
(1) 0
(2) c
(3) a (4) c4
3. If the normal at (ct1, c/t1) on the hyperbola xy = c2 cuts the hyperbola again at ( ct2, c/ t2), then t1 3 t2 =_____
(1) 2
(2) –2
(3) –1
(4) 1
4. The equation of the conjugate hyperbola of xy + 3x – 4y + 13 = 0 is
(1) (x – 4) (y + 3)
(2) (x – 4) (y + 3) = 0
(3) (x – 4) (y + 3) = 25
(4) xy + x – 4y – 13 = 0
5. The length of the semi-transverse axis of the rectangular hyperbola xy = 32 is (1) 32
(2) 16
(3) 64
(4) 8
6. Length of the principal axis of the hyperbola xy = 32 is (1) 8
(2) 16
(3) 82 (4) 2
Answer Key
(1) 2 (2) 1 (3) 3 (4) 3
(5) 4 (6) 2
CHAPTER REVIEW
1. Centre (C)
2. Vertices
3. Foci
1 , SS
4.
5. Ends of latus recta
6. Equation of Transverse axis
7. Equation of
8. Equations of latus recta
9. Equations of directrices
10. Length of transverse axis
11. Length of
axis
12. Length of latus rectum
13. Eccentricity (e)
14. Difference of focal distances (focal radii) of a point P on the hyperbola
15. Distance between the foci
16. Distance between vertices
17. Distance between directrices
18. Parametric points () sec,tanab
tan,secab
1. 2 2 22 1 xy ab −= and 2 2 22 1 xy ab −=− are called
conjugate hyperbolas to each other.
2. If e,e12 are the eccentricities of two conjugate hyperbolas, then 2222 1212 eeee. +=
3. In a hyperbola, if the length of the transverse axis () 2a is equal to the length of the
conjugate axis () 2b then the hyperbola is called rectangular hyperbola
4. A point () 11 , xy is said to be
i. an external point to the hyperbola
0 S = if 11 0 S <
ii. an internal point to the hyperbola
0 S = if 11 0 S >
iii. lies on the hyperbola 0 S = if 11 0. S =
5. Two tangents can be dawn to a hyperbola from an external point.
6. The equation of the tangent to the hyperbola 0 S = at () 11 , Pxy is 1 0 S =
7. The equation of the normal to the hyperbola
2 2 22 1 xy ab −= at ()Px,y11 is
2 2 22 11 axbyab xy +=+
8. The condition that the line ymxc =+ may be a tangent to the hyperbola
2 2 22 1 xy ab −= is 2222 camb =− and the point of contact is 22 ,. amb cc
9. The condition that the line 0. lxmyn++= may be a tangent to the hyperbola
2 2 22 1 xy ab −= is 22222 albmn −= and the point of contact is 22 ,. albm nn
10. The equation of a tangent to the hyperbola
2 2 22 1 xy ab −= may be taken as 222 ymxamb =±−
11. If 12 , mm are the slopes of the tangents through P to the hyperbola 2 2 22 1 xy ab −= then 22 11 1 12 12 2222 11 2 ;. xyyb mmmm xaxa + +==
12. If θ is the angle between the tangents drawn from a point () 11 , xy to the hyperbola
2 2 22 10, Sxy ab =−−= then 11 2222 11 2 tan abS xyab θ= +−+
13. The equation to the director circle of 2 2 22 1 xy ab −= is 2222 xyab +=−
14. The equation to the auxiliary circle of 2 2 22 1 xy ab −= is 222 . xya +=
15. The equation to the chord of contact of () 11 , Pxy with respect to the hyperbola 0 S = is 1 0 S =
16. The equation of the chord of the hyperbola 0 S = having () 11 , Pxy as its mid point is 111SS =
17. The midpoint of the chord 0 lxmyn++= of the hyperbola 2 2 22 1 xy ab −= is 22 22222222 ln , abmn albmalbm
18. The equation to the pair of tangents to the hyperbola S0 = from ()Px,y11 is 2 111 . SSS =
19. The equations sec;tanxayb=θ=θ are called parametric equations of the hyperbola 2 2 22 1 xy ab −= and the point () sec,tanabθθ is called parametric point. It is denoted by () P. θ
20. If ()() 11,sec,tanPxyab=θθ is a point on hyperbola 2 2 22 1 xy ab −= and its foci are S,S1 then 1 sec SPexaaea =−=θ− and 1 1 sec.SPexaaea =+=θ+
21. The equation of the chord joining two points α and β on the hyperbola 2 2 22 1 xy ab −= is cossincos 222 xy ab
22. If α and β are the ends of a focal chord of a hyperbola S = 0, then (i) coscos 22 e α−βα+β =± (ii) tantan()11 2211 ee or ee αβ−+
23. The equation of the tangent at () P θ on the hyperbola 0 S = is sectan1xy ab θ−θ=
24. The tangents at the points ()() , PQαβ intersect at the point is cossin 22 , coscos 22 ab α−βα+β
25. The equation of the normal at () P θ on the hyperbola 0 S = is 22 sectan axbyab +=+ θθ
26. The condition that the line 0 lxmyn++= to be a normal to the hyperbola
2 2 22 1 xy ab −= is ()2 22 22 222 + −= abab lmn
27. The slope form of normal to the hyperbola 2 2 22 1 xy ab −= is () 22 222 mab ymx abm + =±
28. Atmost four normals can be drawn from a point to a hyperbola
29. If 1234 ,,, θθθθ are feet of normals to the hyperbola, then () 1234 21 n θ+θ+θ+θ=+π
30. Let S and S ′ be two foci of hyperbola. The tangent at P to the Hyperbola is internal angular bisector of SPS ∠′ and normal at P to the Hyperbola is external angular bisector of SPS ∠′ .
31. If an incoming light ray passing through one focus and strike convex side of the hyperbola then it will get reflected toward other focus S ′
32. The tangents of the hyperbola which touch the hyperbola at the points at infinity are called asymptotes of the hyperbola.
33. The equations of the asymptotes to the hyperbola 2 2 22 1 xy ab −= are 0,0yyxx abab −=+= and the combined equation of these two asymptotes is 2 2 22 0 xy ab −=
34. The angle between the two asymptotes is 1 2tan b a
or 2 sec–1 e.
35. The product of the perpendiculars from any point on the hyperbola to its asymptotes is 22 22 ab ab +
36. Asymptotes of a hyperbola and its conjugate hyperbola are same.
37. If H , C and A are the equations of a hyperbola, its conjugate hyperbola and its pair of asymptotes respectively, then H + C = 2A
38. The bisectors of the angles between the asymptotes are the transverse, conjugate axes.
39. The parametric equations of 2 xyc = are ;. c xctyt ==
40. Equation of tangent at , c ct t to the hyperbola is 2 x tytc +=
41. The tangent at 1t and 1t to the rectangularhyperbola intersect at 12 1212 2 2 , cttc tttt
42. The equation of normal to at () 11 , xy to the rectangular hyperbola is 22 1111 xxyyxy −=−
43. The equation of normal at , c ct t to the rectangular hyperbola 2 xyc = is 34 0 xtytctc−−+=
JEE MAIN LEVEL
LEVEL-I
Different Forms of Hyperbola
Single Option Correct MCQs
1. The foci of the ellipse 2 2 2 1 16 xy b += and the hyperbola 2 2 1 1448125 xy −= coincide. Then the value of b2 is
(1) 5 (2) 7 (3) 9 (4) 1
2. Equations of directrices of 4x2 – 9y2 = 36 is
(1) 133 x =± (2) 139 x =±
(3) 132 x =± (4) 134 x =±
3. The equations of the axes of the hyperbola 9x2 – 16y2 + 72x – 32y –16 = 0 are
(1) y + 1 = 0, x + 4 = 0
(2) y + 2 = 0, x + 3 = 0
(3) y – 1 = 0, x – 4 = 0
(4) y + 3 = 0, x – 4 = 0
4. If the eccentricity of a hyperbola is 3 , then the eccentricity of its conjugate hyperbola is (1) 2 (2) 3 (3) 3 2 (4) 23
5. The eccentricity of the conjugate hyperbola of the hyperbola x2 – 3y2 = 1 is
(1) 2 (2) 2 3 (3) 4 (4) 4 3
6. The distance between the foci is 4 13 and the length of conjugate axis is 8. Then the eccentricity of the hyperbola is (1) 13 3 (2) 13 5 (3) 13 7 (4) none
7. The latusrectum of a hyperbola 2 2 1 16 xy p −= is 1 4. 2 Its eccentricity e = (1) 4 5 (2) 5 4 (3) 3 4 (4) 4 3
8. If the latus rectum of a hyperbola through one focus subtends 60° at the other focus then its eccentricity is (1) 2 (2) 3 (3) 5 (4) 6
9. If the latus rectum of a hyperbola forms an equilateral triangle with the centre of the hyperbola, then its eccentricity is
(1) 51 2 + (2) 111 2 + (3) 131 23 + (4) 131 23
10. The equations of the transverse and conjugate axes of a hyperbola are respectively x + 2 y – 3 = 0, 2 x – y + 4 = 0 and their respective lengths are 2 and 2 3 . The equation of the hyperbola is
(1) ()() 2322 24231 55xyxy −+−+−=
(2) ()() 2322 23231 55xyxy −+−+−=
(3) 2(2x – y + 4)2 – 3(x + 2y – 3)2 = 1
(4) ()() 22 3 223241 5 xyxy +−−−+=
11. If (5, 12), (24, 7) are the foci of the hyperbola passing through origin, then its eccentricity is
(1) 13 5 (2) 386 13
(3) 386 25 (4) 386 12
12. If a hyperbola has one focus at the origin and its eccentricity is 2 One of the directrices is x + y + 1 = 0. Then the centre of the hyperbola is (1) (–1, –1) (2) (1, –1) (3) (–2, –1) (4) (2, 2)
13. A = (1, 2), B = (6, –10) locus of P such that PA – PB = 13 is (1) a hyperbola (2) a part of a hyperbola (3) a ray with vertex at B not containing A (4) a line segment
14. The locus of the point of intersection of the lines, 2420 xyk−+= and 2420 kxky+−= ( k is any non–zero real parameter), is
(1) an ellise whose eccentricity is 1 3
(2) an ellipse with length of its major axis 82
(3) a hyperbola whose eccentricity is 3 (4) a hyperbola with length of its transverse axis 82
15. If e1, e2, e3 are eccentricities of a parabola, ellipse and hyperbola respectively and 222222 123123 46 44 , 99 eeeeee ++=−+= then the eccentricity of conjugate hyperbola is (1) 2 (2) 2 3 (3) 4 3 (4) 2
16. The equation ()()() 222 2 212 xyxy −+−−++ = c will represent a hyperbola if (1) c ∈ (0, 6) (2) c ∈ (0, 5) (3) () 0, 17 c ∈ (4) c ∈ R
17. The eccentricity of the conic 4(2y– x – 3)2 – 9(2x + y – 1)2 = 80 is (1) 3 13 (2) 13 3 (3) 13 (4) 3
18. If (3, 1) is a focus and x = 0 is the corresponding directrix of a conic with eccentricity 2, then its vertices are (1) (1, 1) (–3, 1) (2) (2, 1) (–3, 1) (3) (1, 1) ( 6, 1) (4) (1, –1) ( 1, –3)
19. The equation of a hyperbola, conjugate to the hyperbola 2x2 + 3xy –2y2 + 3x + y + 2 = 0 is 2x2 + 3xy – 2y2 + 3x + y + k = 0 then k = (1) 0 (2) 1 3) –4 (4) 4
20. The sides AC and AB of a DABC touch the conjugate hyperbola of the hyperbola
2 2 22 1. xy ab −= If the vertex A lies on the ellipse
2 2 22 1, xy ab += then the side BC must touch (1) parabola (2) circle (3) hyperbola (4) ellipse
21. If S 1 and S 2 are the foci of the hyperbola whose transverse axis length is 4 and conjugate axis length is 6, S3 and S4 are the foci of the conjugate hyperbola, then the area of the quadrilateral S1S2S3S4 is (1) 24 (2) 26 (3) 22 (4) 12
Numerical Value Questions
22. Let the latus rectum of the hyperbola 22 2 1 9 xy b −= subtend an angle 3 π at the center of the hyperbola. If b 2 is equal to () 1, l n m + where l and m are co-prime numbers, then l2+m2+n2 is ______
23. If e is the eccentricity of the hyperbola (5 x–10)2+(5y+15)2 = (12x–5y+1)2 then 25 13 e is equal to _____
Tangent and Normal to Hyperbola
Single Option Correct MCQs
24. Equation of one of the tangents passing through (2, 8) to the hyperbola 5 x2 – y2 = 5 is
(1) 3x + y – 14 = 0 (2) 3x – y + 2 = 0
(3) x + y + 3 = 0 (4) x – y + 6 =0
25. A tangent to the curve 9b2x2 – 4a2y2 = 36a2b2 makes intercepts of unit length on each of the coordinate axes, then the point (a, b) lies on
(1) x2 – y2 = 1
(2) x2 + y2 = 1
(3) 4x2 – 9y2 = 1
(4) 4x2 + 9y2 = 1
26. The point of contact of 9x+8y–11=0 to the hyperbola 3x2 – 4y2 = 11 is
(1) (3, –2) (2) (3, 2) (3) (–3, –3) (4) (3, 3)
27. The equations of the tangents to the hyperbola 3x2 – 4y2 = 12 which are parallel to the line 2x + y + 7 = 0 are
29. Let P (6, 3) be a point on the hyperbola 2 2 22 1. xy ab −= If the normal at the point P intersects the x –axis at (9, 0), then the eccentricity of the hyperbola = (1) 5 2 (2) 3 2 (3) 2 (4) 3
30. If x = 9 is a chord of contact of the hyperbola x2 – y2 = 9, then the equation of the tangent at one of the points of contact is (1) 3 2 0 xy++= (2) 3–2 2 –3 0 xy = (3) 3–2 6 0 xy += (4) – 3 2 0 xy +=
31. The coordinates of a point on the hyperbola, 2 2 1 2418 xy −= , which is nearest to the line 3x + 2y + 1 = 0 are
(1) (6, 3) (2) (–6, –3) (3) (6, –3) (4) (–6, 3)
Numerical Value Questions
32. If the tangents drawn from the point ( p , 2) to the hyperbola 22 1 169 xy −= are perpendicular, then the value of p 2 is
33. Number of integral values of b for which the tangent parallel to line y = x+1 that can be drawn to the hyperbola 22 2 1 5 xy b −= is
Parametric Equations
Single Option Correct MCQs
34. A tangent drawn to hyperbola 2 2 22 1 xy ab −= at 6 P π
forms a triangle of area 3a2 square units, with coordinate axes, then the square of its eccentricity is (1) 15 (2) 24 (3) 17 (4) 14
35. If the intercepts made by tangent, normal to a rectangular hyperbola x2 – y2 = a2 with x–axis are a1, a2 and with y–axis are b1, b2 then a1a2 + b1b2 = (1) 0 (2) 1 (3) –1 (4) a2
Pair of Tangents
Single Option Correct MCQs
36. Given below are the two statements
Assertion(A) : There are infinite points from which two mutually perpendicular tangents can be drawn to the hyperbola 2 2 1 916 xy −=
Reason (R) : The locus of point of intersection of perpendicular tangents lies on the ellipse.
In light of the above statements, choose the correct answer from the options given below.
(1) Both (A) and (R) are true and (R) is the correct explanation of (A)
(2) Both (A) and (R) are true but (R) is not the correct explanation of (A)
(3) (A) is true but (R) is false
(4) Both (A) and (R) are false
Numerical Value Questions
37. If distance between to parallel tangents having slope m drawn to 22 1 949 xy −= is 2 then value of |m| is ____
Asymptotes
Single Option Correct MCQs
38. The equation of a hyperbola, conjugate to the hyperbola 2x2 + 3xy – 2y2 + 3x + y + 2 = 0 is 2x2 + 3xy – 2y2 + 3x + y + k = 0. Then k =
(1) 0 (2) 1 (3) –4 (4) 4
39. S = 0 is a hyperbola. If S + k = 0 (k is a real number) represents equation of the asymptotes, then S+ 2k = 0 represents (1) Hyperbola (2) Ellipse (3) Parabola (4) Circle
40. The line x + y + 1 = 0 is an asymptote of x2 – y2 + x – y – 2 = 0. The other asymptote is
(1) x + y = 0 (2) x – y = 0
(3) x – y = 1 (4) x – y + 1 = 0
41. If the equation of one asymptote of the hyperbola 14x2 + 38xy + 20y2 + x – 7y – 91 = 0 is 7x + 5y – 3 = 0 then the other asymptote is
42. Area of the triangle formed by any tangent to the hyperbola 2 2 22 1 xy ab −= with its asymptotes is (1) ab (2) abc (3) 4 ab (4) a2b2
43. If H ( x, y ) = 0 represents the equation of a hyperbola and A ( x, y ) = 0, C ( x, y ) = 0 the joint equation of its asymptotes and the conjugate hyperbola respectively, then for any point ( a,b ) in the plane H ( a, b ), A(a, b) and C(a, b) are in (1) AP (2) GP (3) HP (4) AGP
Numerical Value Questions
44. The angle between the asymptotes of the hyperbola 5 x 2–2 7 xy – y 2–2 x +1=0 is k π , then k =_____
Rectangular Hyperbola
Single Option Correct MCQs
45. If (xi, yi), i = 1, 2, 3, 4 are concyclic points on xy = 4, then x1x2x3x4 = (1) 4 (2) –4 (3) –16 (4) 16
46. If f ( x ) = x 3 + αx 2 + β x + γ where α , β, γ are rational numbers and two roots of f(x) = 0 are eccentricities of a parabola and a rectangular hyperbola then α + β + γ is equal to = (1) –1 (2) 0 3) 1 (4) 2
47. Given below are the two statements .
Statement-I : If a circle S = 0 intersects a hyperbola xy = 4 at four points. Three of them are (2, 2), (4, 1) and (6, 2 3 ), then coor-dinates of the fourth point are ( 1 4 , 16)
Statement–II : If a csircle S = 0 intersects a hyperbola xy = c2 at t1, t2, t3, t4 then t1t2t3t4 = 1.
In light of the above statements, choose the correct answer from the options given below.
(1) Both the statements are True and II is the correct explanation of I
(2) Both the statements are True but II is not the correct explanation of I
(3) Statement–I is True and Statement–II is False
(4) Statement–I is False and Statement–II is True
48. Area of the triangle formed by a tangent to the hyperbola xy = 75 and its asymptotes
(1) 25(unit)2
(2) 50(unit)2
(3) 100(unit)2
(4) 150(unit)2
49. Equation of the hyperbola through the point
(1, –1) and having asymptotes x + 2y + 3 = 0 and 3x + 4y + 5 = 0 is
(1) 3x2 – 10xy + 8y2 – 14x + 22y + 7 = 0
(2) 3x2 + 10xy + 8y2 – 14x + 22y + 7 = 0
(3) 3x2 – 10xy – 8y2 + 14x + 22y + 7 = 0
(4) 3x2 + 10xy + 8y2 + 14x + 22y + 7 = 0
50. The combined equation of the asymptotes of the hyperbola xy + x + y + 5 = 0 is
(1) xy = 0
(2) (x – 1)(y – 1) = 0
(3) (x + 1)(y – 1) = 0
(4) (x + 1)(y + 1) = 0
51. If a hyperbola has on focus at the origin and its eccentricity is 2 one of the directrices is x + y + 1 = 0. Then equations to its asymptotes are
(1) x = 1 , y = 1
(2) x + 1 = 0 , y + 1 = 0
(3) x = 3 , y = 3
(4) x + 3 = 0 , y + 3 = 0
52. The points of intersection of asymptotes of a hyperbola with directrices lies on
(1) Director circle
(2) Auxiliary circle
(3) Circle on SS1 diameter
(4) None
Numerical Value Questions
53. If a circle cuts a rectangular hyperbola xy = c2 in A, B, C and D and the parameters of these four points be t 1 , t 2 , t 3 and t 4 respectively, then the value of t1.t2.t3.t4 must be______.
54. The eccentricity of the conic represented by x2 – y2 – 4x + 4y +16 = 0 is k, then k2 = _______.
LEVEL-II
Different Forms of Hyperbola
Single Option Correct MCQs
1. A hyperbola passes through the foci of the ellipse 2 2 1 2516 xy+= and its transverse and conjugate axes coincide wit h m ajor and minor axes of the ellipse respectively. If the product of their eccentricities is one,then the equation of the hyperbola is
2. The vertices of a hyperbola H are ( ± 6, 0) and its eccentricity is 5 2 . Let N be the normal to H at a point in the first quadrant and parallel to the line 222 xy+= . If ‘d’ is the length of the line segement of N between H and the y-axis then ‘d2’ is equal to ______
Tangent and Normal to Hyperbola
Single Option Correct MCQs
3. A normal to the hyperbola 4x2 – 9y2 = 36 meets the coordinate axes x and y at A and B respectively. If the parallelogram OABP (O is origin) is formed, then the locus of P is
(1) 4x2 – 9y2 = 121
(2) 4x2 + 9y2 = 121
(3) 9x2 – 4y2 = 169
(4) 9x2 + 4y2 = 169
4. The locus of the point of intersection of the tangents at the ends of normal chord of the hyperbola x2 – y2 = a2 is
tangent at P to the hyperbola meets the straight lines bx – ay = 0 and bx + ay = 0 respectively at Q and R then CQ.CR = (1) a2 – b2 (2) a2 + b2 (3) 22 11 ab (4) 22 11 ab +
6. A hyperbola passes through the point () 2, 3 P and has foci at (±2, 0). Then the tangent to this hyperbola at P also passes through the point
7. Tangents are drawn from the point (α, b) to the hyperbola 3x2–2y2 = 6 and are inclined at angle θ and φ to the x -axis. If (tan θ ) (tan φ ) = 2, then the value of 22 2 7 αβ is
8. If area of the triangle formed by latus rectum and tangents at the end points of latus rectum of 22 1 169 xy −= is A, then 80 A is ______
Parametric Equations
Single Option Correct MCQs
9. Let A (2secθ, 3tanθ) and B (2secψ, 3tanψ) where θ + ψ = 2 π , be two poin ts on the hyperbola 2 2 1. 49 xy −= If ( α , β) is point of intersection of normals to the hyperbola at A and B, then β = (1) 13 3 (2) 13 3 (3) 3 13 (4) 3 13
10. If a ray of light incident along the line () 354215 xy+−= gets reflected from the hyperbola 2 2 1, 169 xy −= then reflected ray goes along the line (1) 250xy−+= (2) 250 yx−+=
(3) 250 yx−−= (4) () 3425150 xy−++=
Pair of Tangents
Single Option Correct MCQs
11. The locus of the point of intersection of tangents to the hyperbola x2 – y2 = a2 which include an angle of 45º is (1) (x2 + y2)2 = 4a2(x2 + y2 + a2)
12. Tangents are drawn from any point on the hyperbola 2 2 1 94 xy −= to the circle x2 + y2 = 9. If the locus of the mid–point of the chord of contact is a(x2 + y2)2 = bx2 – cy2 , then the value of a2 + b2 + c2= (1) 7863 (2) 7853 (3) 7873 (4) 8763
13. The locus a point P ( α , β) moving under the condition that the line y = αx + β is a tangent to the hyperbola 2 2 22 1 xy ab −= is (1) a parabola (2) an ellipse (3) a hyperbola (4) a circle
Asymptotes
Single Option Correct MCQs
14. If 2x2 + 5xy + 2y2 – 11x – 7y + k = 0 is the pair of asymptotes of the hyperbola 2 x2 + 5xy +2y2 – 11x – 7y – 4 = 0 then k = (1) 3 (2) 4 (3) 5 (4) 6
15. The area (in square units) of the equilateral triangle formed by the tan gent at () 3,0 to the hyperbola x2 – 3y2 = 3 with the pair of asymptotes of the hyperbola is (1) 2 (2) 3 (3) 1 3 (4) 23
16. If foci of hyperbola lie on y = x and one of asymptote is y = 2 x , then equation of hyperbola, given that it passes through (3, 4) is
(1) 22 5 50 2 xyxy+−+=
(2) 2x2 – 2y2 + 5xy + 5 = 0
(3) 4x2 + 4y2 + 5xy + 10 = 0
(4) No sufficient data
Numerical Value Questions
17. A hyperbola passes through (2, 3) and has asymptotes 3x–4y+5 = 0 and 12x+5y–40 = 0 and the equation of its transverse axis is ax+b y = 265 then 2 ba is _____
Rectangular
Hyperbola
Single Option Correct MCQs
18. If a variable circle x 2 + y 2 – 2 ax + 4 ay = 0 intersect the hyperbola xy = 4 at the points (xi,yi) i =1, 2, 3, 4 then the locus of the point
12341234 , 44 xxxxyyyy ++++++
(1) y + 2x = 0
(2) x – 2x + 5 = 0
(3) y – 2x = 0
(4) y + 4x – 7 = 0
is
19. The line x + y + 1 = 0 is an asymptote of x2 – y2 + x – y – 2 = 0. The other asymptote is
(1) x + y = 0
(2) x – y = 0
(3) x – y = 1
(4) x – y + 1 = 0
20. The equation of a line passing through the centre of a rectangular hyperbola is x – y –1=0 . If one of the asymptotes is 3x – 4y – 6 = 0, the equation of the other asymptote is
(1) 4x + 3y + 17 = 0
(2) 4x – 3y + 17 = 0
(3) –4x – 3y + 17 = 0
(4) –4x + 3y + 1 = 0
21. The centre of a rectangular hyperbola lies on the line y = 2x. If one of the asymptotes is x + y + c = 0, then the other asymptote is
(1) x – y – 3c = 0
(2) 2x – y + c = 0
(3) x – y – c = 0
(4) 3x–3y–c=0
22. The asymptotes of the hyperbola
2 2 22 1 xy ab −= form with any tangent to the hyperbola a triangle whose area is a2tanλ in magnitude, then its eccentricity is (1) secλ (2) cosecλ (3) sec2λ (4) cosec2λ
Numerical Value Questions
23. If the normal to the rectangular hyperbola xy = c2 at the point , c ct t meets the curve again at ',, ' c ct t then |t3 . t’| is____.
24. Let H: xy–y–2x–2 = 0. If normal drawn to H at P(3, 4) intersects the curve again at Q(α+1, b +1), then α β is _____
25. Let x2+y2 = 4r2 and xy = 1 intersects at A and B in first quadrant. If AB = 28 , units, then the value of |r| is ____
LEVEL-III
Single Option Correct MCQs
1. The lines of the form x cosϕ + y sinϕ = p are chords of the hyperbola 4 x2 – y2 = 4a2 which subtend a right angle at the centre of the hyperbola. If these chords touch a circle with centre at (0, 0), then the radius of that circle is
2. If tangents to the parabola y2 = 4ax intersects the hyperbola 2 2 22 1 xy ab −= at A and B, then the locus of point of intersection of tangents at A and B is parabola y2 = kx, then k = (1)
3. Given below are the two statements. One is labelled Assertion (A) and the other is labelled Reason (R)
Assertion (A): Ellipse 2 2 1 2516 xy+= and 12 x 2 – 4 y 2 = 27 intersect each other at right angle
Reason (R) : Given ellipse and hyperbola have same foci
In light of the above statements, choose the correct answer from the options given below.
(1) Both (A) and (R) are true and (R) is the correct explanation of (A)
(2) Both (A) and (R) are true but (R) is not the correct explanation of (A)
(3) (A) is true but (R) is false
(4) Both (A) and (R) are false
4. A tangent to the circle x2 + y2 = 4 intersects the hyperbola x2 – 2y2 = 2 at P and Q. If locus of mid–point of PQ is (x2 – 2y2)2 = L(x2 + 4y2); then L =
(1) 2 (2) 4 (3) 6 (4) 8
5. Area of quadrilateral formed with the foci of the hyperbola 2 2 22 1 xy ab −= and 2 2 22 1 xy ab −=− is
(1) 4(a2 + b2)
(2) 2(a2 + b2)
(3) (a2 + b2)
(4) () 22 1 2 ab +
6. The locus of the centriod of the triangle formed by any point P on the hyperbola 16x2 – 9y2 + 32x + 36y – 164 = 0 and its foci is
(1) 9x2 – 16y2 + 36x + 32y – 144 = 0
(2) 16x2 – 9y2 + 32x + 36y – 36 = 0
(3) 9x2 – 16y2 + 36x + 32y – 36 = 0
(4) 16x2 – 9y2 + 32x + 36y – 144 = 0
7. If a hyperbola passing through the origin has 3x – 4y –1 = 0 and 4x – 3y –6 = 0 as its asymptotes, then the equation of its transverse and conjugate axes are
(1) x – y – 5 = 0 and x + y + 1 = 0
(2) x – y = 0 and x + y + 5 = 0
(3) x + y – 5 = 0 and x – y – 1 = 0
(4) x + y – 1 = 0 and x – y
8. If angle between the asympto tes of the hyperbola 2 2 22 1 xy ab −= is 60° and product of perpendiculars drawn from foci to any tangent is 9, then the locus of point of intersection of perpendicular tangents of the hyperbola can be
(1) x2 + y2 = 6
(2) x2 + y2 = 9
(3) x2 + y2 = 3
(4) x2 + y2 = 18
9. Let any double ordinate PNP ’ of the hyperbola 2 2 1 2516 xy −= be p rodu ced on both sides to meet the asymptotes in Q and Q’. Then PQ .PQ’ is equal to (1) 25
(2) 16
(3) 41
(4) none of these
10. If (x – 1)(y – 2) = 5 and (x – 1)2(y + 2)2 = r2 intersect at four points A, B, C, D and if centroid of D ABC lies on line y = 3x – 4, then locus of D is
(1) y = 3x
(2) x2 + y2 + 3x + 1 = 0
(3) 3y = x + 1
(4) y = 3x + 1
Numerical Value Questions
11. Points A(t 1 ),B(t 2 ), C(t 3 ) lie on the rectangular hyperbola xy = 1 and P, Q, R are the points of intersection of tangents at A, B, C taken in pairs. If the ratio of area of the D PQR to the area of the D AB C is () () () 123 122331 , kttt tttttt +++ then k = ____
12. The ord inate of any point P on the hyperbola 25x2–16y2 = 400 is produced to cut the asymptotes in the points Q and R then the value of QP.PR is equal to ____
13. If 5 x 2 + λ y 2 = 20 represents a rectangular hyperbola, then λ2 is equal to_____.
14. The point on the hyperbola 2 2 1 2418 xy −= which is nearest to the line 3x + 2y + 1 = 0 is (h, k), then |h + k| = ______.
15. Let the equation of two diameters of a circle x2 + y2 – 2x + 2fy + 1 =0 be 2px – y = 1 and 2x + py = 2p. Then the slope m ∈ (0, ∞) of the tangent to the hyperbola 3 x 2 – y 2 = 3 passing through the centre of the circle is equal to_____
16. Consider a hyperbola H : x2 – 2y2 = 4. Let the tangent at a point P(4, 6 ), meet the x–axis at Q and latus rectum at R(x1, y1), x1 > 0. If F is a focus of H which is nearer to the point P, then the area of DQFR is equal to 7 b a then a + b =_______.
17. If slope of the chord of hyperbola 2x2 – 3y2 = 6 which is bisected at (5, 3) is p q , where p and q are co-prime positive integers, then the value of |p – q| is______.
THEORY-BASED QUESTIONS
Very Short Answer Questions
1. If the normal at ()Pasec,btanθθ to the hyperbola 22 22 xy1 ab −= meets the transverse axis in G, then what is the minimum length of PG ?
2. If a, b are eccentricities of a hyperbola and its conjugate hyperbola, then what is the value of 22 22 ab ab + ?
3. What is the eccentricity of the locus of the point tttt eeee , 22 +−
?
4. If α and β are two points on the hyperbola 2 2 22 y x 1 ab −= and the chord joining these two points passes through the focus ()ae,0, then what is the value of ecos 2 α−β ?
5. If the normal at 1 1 c ct, t on the hyperbola 2 xyc = cuts the hyperbola again at 2 2 c ct, t , then what is the value of 3 12 tt?
6. If ()()() 112233 Px,y,Qx,y,Rx,y and ()Sx,y44 are four concylic points on the rectangular hyperbola 2 xyc = , then what is the orthocentre of PQR ∆ ?
7. What is the equation of the chord joining two points () x,y11 and () x,y22 on the rectangular hyperbola 2 xyc = ?
8. If the intercepts made by tangent, normal to a rectangular 222 xya −= with x-axis are a,a12 and with y-axis are b,b12 , then what is the value of 1212 aabb + ?
9. If the line xmy1 += is a normal to the hyperbola 2 2 22 y x 1, ab −= then what is the value of 22 22 ab m ?
10. What is the angle between the asymptotes of the hyperbola 222 xya? −=
Statement Typed Questions
Each question has two statements: statement I (S-I)and statement II (S-II). Mark the correct answer as
(1) if both statement I and statement II are correct
(2) if both statement I and statement II are incorrect
(3) if statement I is correct, but statement II is incorrect
(4) if statement I is incorrect, but statement II is correct
11. S - I : Two circles are given such that they neither intersect nor touch. Then locus of the centre of a variable circle which touches both the circles externally is a hyperbola.
S - II : A normal to a hyperbola other than transverse axis never pass through the focus.
12. S - I : The product of perpendiculars from foci of a hyperbola to any tangent is 2b
S - II : The product of the perpendiculars from any point on the hyperbola to its asymptotes is 22 22 ab . ab +
13. S - I : If ()() asec,btan,asec,btan ααββ are the ends of a focal chord, then 2 2 1e tantan 22 1e αβ− = +
S - II : If the angle between the asymptotes of a hyperbola is 2θ , then esec=θ
14. S - I : The area of triangle formed by any tangent to the hyperbola with its 2 2 22 y x 1 ab −= asymptotes is ab.
S - II : Hyperbolas is 2 222 xycandxya =−= intersect orthogonally.
15. S-I : If the differential equation dy3y dx2x = represents a hyperbola with eccentricity 2
S - II : Eccentricity of the hyperbolas ()()()() 2222 x3y2x1y11 −+−−+++= is 5
16. S-I : Let () 2 2 2 y x Sx,yR: 1,wherer1. 1r1r =∈−=≠± +−
Then S represents A hyperbola whose eccentricity is 2 ,where0r1. r1 << +
S - II : The locus of the point of intersection of the lines, 2xy42k0and2kxky420 −+=+−= (k is any non zero real parameter), is A hyperbola whose eccentricity is 3 2
17. S-I : If the normal at 1 1 c ct, t
on the hyperbola 2 xyc = cuts the hyperbola again at 2 2 c ct, t , then 3 12 tt1 =−
S - II : If two distinct tangents can be drawn from the point () ,2 α on different branches of the hyperbola 2 2 y x 1 916 −= , then 3 2 α<
18. S-I : If values of m for which the line ymx25 =+ touches the hyperbola
22 16x9y144 −= are roots of the equation () 2 xabx40 −+−= , then value of () ab + is equal to 0.
S - II : If S and T are foci of 2 2 y x 10 169 −+= If P is a point on the hyperbola, then SPPT8 −=
19. S-I : If m is a variable, the locus of the point of the point of intersection of the lines y x m 32 −= and y x1 32m += is a hyperbola.
S-II : The angle between the asymptotes of the hyperbola 2 2 22 y x 1 ab −= is () 1 2tane
20. S - I : If two points P and Q on the hyperbola 2 2 22 y x 1 ab −= , which centre C be such that CP is perpendicular to CQ, then 22 11 CPCQ + is equal to 22 11 ab +
S - II : If the line xmy1 += is a normal to the hyperbola 2 2 22 y x 1 ab −= then () 22 2 22 22 ab ab m −=−
Assertion and Reason Questions
In each of the following questions, a statement of Assertion (A) is given, followed by a corresponding statement of Reason (R). Mark the correct answer as
(1) if both (A) and (R) are true and (R) is the correct explanation of (A)
(2) if both (A) and (R) are true but (R) is not the correct explanation of (A)
(3) if (A) is true but (R) is false
(4) if both (A) and (R) are false
21. (A) : A hyperbola and its conjugate hyperbola have same asymptotes.
(R) : A hyperbola and its pair of asymptotes differ by a constant only.
22. (A) : Latus rectum of the hyperbola 222 xya −= is equal to length of transverse axis.
(R) : In a hyperbola lengths of latus rectum and transverse axis are equal.
23. (A) : Maximum distance from origin to a normal to the hyperbola
2 2 22 y x 1 ab −= is ab .
(R) : The minimum value of 2222 asecbcosecθ+θ is ab +
24. (A) : If the line y3x=+λ touches the hyperbola 22 9x5y45, −= then
2 54 λ=
(R) : the line ymxc =+ touches the hyperbola, if 2222 camb =+
25. (A) : Let () 2 2 2 y x Sx,yR: 1,wherer1. 1r1r
Then S represents A hyperbola whose eccentricity is 2 ,where0r1. r1 << +
(R) : The locus of the point of intersection of the lines, 2xy42k0and2kxky420 −+=+−=
(k is any non zero real parameter), is A hyperbola whose eccentricity is 3 2
26. (A) : If a rectangular hyperbola () () x1y24 −−= cuts a circle 22 xy2gx2fyc0 ++++= at points ()()() 3,4,5,3,2,6 and () 1,0 , then the center of the circle is 79 , 22
(R ) : If a circle cuts a rectangular hyperbola, then arithmetic mean of points of intersections is the mid-
point of centres of hyperbola and circle.
27. (A) : the point a1b1 t,t 2t2t
lies on the hyperbola for all values of ()tt0 ≠
(R) : Locus of point a1b1 t,t 2t2t
is a hyperbola if {}tR0 ∈−
28. (A) : If PQ is a double ordinate of the hyperbola 2 2 22 y x 1 ab −= such that OPQ is an equilateral triangle, O being the center of the hyperbola then the range of the eccentricity of the hyperbola is 2 e 3 >
(R) : I f e is the eccentricity of the hyperbola 2 2 22 y x 1 ab −= and θ is angle between the asymptotes, then cos 2 θ
is equal to 1 e
29. (A) : The portion of the tangent to the hyperbola xy = c 2 intercepted between the asymptotes is bisected at the point of contact.
(R) : Area of triangle formed by the tangent to the hyperbola xy = c2 and asymptotes is 2c2
30. (A) : If the circle 222xya += intersects the hyperbola 2 xyc = in four points 1234 xxxx0 +++=
(R) : If the circle 222xya += intersects the hyperbola 2 xyc = in four points 4 1234 xxxxc =
JEE ADVANCED LEVEL
Multiple Option Correct MCQs
1. If the circle x 2 + y 2 = a 2 intersects the hyperbola xy = c2 in four points P(x1, y1), Q(x2, y2), R(x3, y3), S(x4, y4), then
(1) x1 + x2 + x3 + x4 = 0
(2) y1 + y2 + y3 + y4 = 0
(3) x1 x2 x3 x4 = c4
(4) y1 y2 y3 y4 = c4
2. If the normal at P to the rectangular hyperbola x 2 – y 2 = 4 meets the axes in G and g and C is the centre of the hyperbola, then
(1) PG = PC (2) Pg = PC
(3) PG = Pg (4) Gg = 2PC
3. If the line ax + by + c = 0 is a normal to the hyperbola xy = 1, then
(1) a > 0, b > 0 (2) a > 0, b < 0
(3) a < 0, b > 0 (4) a < 0, b < 0
4. The coordinates of a point common to a directrix and an asymptote of the hyperbola
2 2 1 2516 xy −= are
(1) 2520 , 4141
(3) 2520 , 33
(2) 2520 , 4141
(4) 2520 , 33
5. The tangent to the hyperbola x2 – 3y2 = 3 at the point () 3,0 when associated with two asymptotes constitutes: (1) right angle isosceles triangle (2) an equilateral triangle (3) a triangle whose area 3 (4) a right isosceles triangle
6. e be the eccentricity of a hyperbola and f ( e ) be the eccentricity of its conjugate hyperbola and (f o f o f o . of n times) ( e ) is
F(e) Then
(1) 4 if n is even
(2) 4 if n is odd
(3) 22 if n is even
(4) 22 if n is odd
7. A straight line touches the rectangular hyperbola 9 x 2 – 9 y 2 = 8 and the parabola y2 = 32x. An equation of the line is
(1) 9x +3y – 8 = 0 (2) 9x –3y + 8 = 0
(3) 9x +3y + 8 = 0 (4) 9x –3y – 8 = 0
8. The differential equation 3 2 dxy dyx = represents a family hyperb ola s with (exceptional to pair of lines)
(1) 3 2 (2) 5 3 (3) 2 (4) 5 2
9. On the xy – plane, the path defined by the equation () 11 0, mmn k xyxy ++= + is
(1) a parabola if m = 1 2 , k = –1, n =0
(2) a hyperbola if m–1, k = –1, n = 0
(3) a pair of line if m = k = n = 1
(4) a pair of line if m = k = –1, n = 1
10. If log4(x + 2y) + log4(x – 2y) = 1 is satisfied by a set points P(x, y), then
(1) locus of P is a branch of hyperbola
(2) locus of P consists of two branches of hyperbola
(3) rccentricity of the curve is 5 2
(4) the minimum value of |x| – |y| equals 3
11. If the curve C1 : xy = 4 cuts the curve C2 : x2 + y2 = 17 at two points A(x1, y1) & B(x2, y2) in the 3rd quadrant. Then which of the following is/are true?
(1) orthogonally they intersect
(2) length of AB is 32 units
(3) acute angle between the tangents drawn to C1 at A and B is 1 15 tan 8
(4) acute angle between the tangents drawn to C2 at A and B 1 15 tan 8
= 0 Intersects at two points A a nd B in the first and fourth quadrant respectively. Tangents are drawn at A and B to both C1 and C2 to meet at C and D, then
(1) The quadrilateral ACBD is a cyclic quadrilateral
(2) The quadrilateral ACBD is rhombus
(3) Area of the quadrilateral ACBD is 5 3 units
(4) Area of the quadrilateral ACBD is 1 3 units
13. If P(4K-6, K) is such that two tangents drawn from P to x 2 -8 y 2 = 16, are two different branches of it, then K can be (1) 0 (2) 1 (3) 3 (4) 5
14. Two rays aimed towards the point (5, 0), after reflection from outer surface of the hyperbola 9x2 – 16y2 = 144 at points P and Q intersect at R. If abscissa of points P and Q is 8 , then which of the following options is/are correct?
(1) Area of D PQR is 27 3 sq.units
(2) Area of D PQR is 39 3 sq.units
(3) Angle between asymptotes of the given hyperbola is 1 3 tan 4
(4) Angle between asymptotes of the given hyperbola is 1 24 tan 7
15. If the normals at ( x 1, y 1), ( x 2, y 2), ( x 3, y 3) and( x 4 , y 4 ) on xy = c 2 are concurrent at ( α , b ), then
(1) x1+x2+x3 +x4 = α
(2) y1+y2+y3 +y4 = b
(3) x1x2x3x4 = –c4
(4) y1y2y3y4 = –c4
16. Let a and b be positive real numbers such that a > 1 and b < a. Let P be point in the first quadrant that lies on the hyperbola 2 2 22 1 xy ab −= . Suppose the tangent to the hyperbola at P passes through the point (1, 0), and suppose the normal to the hyperbola at P cuts off equal intercepts on the coordinate axes. Let D denote the area of the triangle formed by the tangent at P, the normal at P and the x - axis. If e denotes the eccentricity of the hyperbola, then which of the following statements is/ are TRUE?
(1) 1< e < 2
(2) 2 < e < 2
(3) D = a4
(4) D = b
17. For hyperbola 2 2 22 1 xy ab −= , let n be the number of points on the plane through which perpendicular tangents are drawn
(1) if n = 1, then e = 2
(2) if n > 1, then 0 < e < 2
(3) if n = 0, then e > 2
(4) none of these
Numerical /Integer Value Questions
18. A circle with centre (3α, 3β) and of variable radius cuts the hyperbola x2 – y2 = 36 at the points P, Q, R and S. If the locus of centroid of D PQR is (x – 2α)2 – (y – 2β)2 = λ, then the value of λ is ______.
19. A circle cuts two perpendicular lines so that each intercept is of given length .The locus of the center of the circle is a conic whose eccentricity is ______.
20. Tangents are to drawn to the hyperbola 4x2 – y2 = 36 at the points P and Q. If these tangents intersect at the point T(0, 3), then the area (in sq. units) of the DPTQ is _____.
21. Let PN be the ordinate of a point P on the hyperbola 2 2 22 1 (97)(79) xy −= and the tangent at P meets the transverse axis in T, O is the origin, then . 2010 ONOT
=______, wh ere [ ] denotes the greatest integer function.
22. If a chord of hyperbola xy = c2 is normal AB at point A subtending an angle α at origin O , then the value of () () sin sin A A α α + where A = ∠ OAB is equal to______.
23. If the pair of conjugate diameters meet the hyperbola 2 2 2516 xy and its conjugate hyperbola in P and D respectively and CP2 – CD 2 = K where C is the centre, then K =_______.
24. If four points be taken on a rectangular hyperbola such that the chord joining any two is perpendicular to the chord joining the other two and if α , β, γ, δ be the inclinations to either asymptote of the straight lines joining these points to the centre, then the vaue of tanαtanβtanγtanδ must be_____.
25. If a hyperbola be rectangular and its equation be xy = c 2 then the locus of the middle points of chords of constant length 2d is (x2 + y2)(xy – c2) = dλ xy, then the value of λ must be______.
26. PP 1 is a diameter of the Rectangular hyperbola xy = c2 , then the locus of point of intersection of the tangents drawn at P with the lines passing through P 1 and parallel to any asymptotes is xy + kc2 = 0, then k = _____.
27. A tansversal cut the same branch of a hyperbola 2 2 22 1 xy ab −= in P, P ’ and the asymptotes in Q, Q ’, then ( PQ + PQ ’) –(P’Q’ + P’Q) is equal to ______.
28. The graphs of x2 + y2 + 6x – 24y + 72 = 0 and x2 – y2 + 6x + 16y – 46 = 0, intersect at 4 points. The sum of distances of these 4 points from point (–3, 2) is k, then 10 k is ______
29. If the four normals at four points (xi, yi) where i = 1,2,3,4 on the hyperbola 2 2 22 1 xy ab −= are concurrent, then
30. If e is eccentricity of a hyperbola whose asymptotes are 3x+4y = 2 and 4x–3y+5 = 0, then e2 = ____
31. C is the centre of x2–4y2 = 4. A is a point on the hyperbola. The tangent at A meets the lines x–2y = 0 and x+2y = 0 at Q and R respectively, then CQ.CR = ____
32. If the product of the length of perpendiculars drawn from any point on the hyperbola x2–2y2–2 = 0 to its asymptotes is _____.
33. Two tangents to the hyperbola 2 2 22 1 xy ab −= having slopes m1 and m2 intersect the axes at four concyclic points. Then the value of m1m2 is ______.
Passage-based Questions
Q. (34 –36)
A circle and a rectangular hyperbola xy = c2 meet in four points ‘ t1’,’t2’, ‘ t3’ and ‘t4’.
34. Then t1t2t3t4 is equal to (1) –1 (2) 1 (3) 0 (4) 2
35. If the sum of the slopes of the normals from point P on given hyperbola is constant k(k >0), then the locus of point P is
(1) x2 = kc2 (2) x2 = kc (3) x2 + kc2 = 0 (4) None of these
36. The mean position of the four points will be
(1) , 22 gf
(3) , 22 gf
Q. (37 –39)
(2) , 22 gf
(4) , 22 gf
If we rotate the axes of the rectangular hyperbola x 2 – y 2 = a 2 throug h an angle 4 π in the clockwise direction, then the equation x 2 – y 2 = a 2 re du ces to
(say). Si nce x = ct , y = c t satisfies xy = c 2 . ()() ,, 0 c xyctt t
is called a t point on the rectangular hyperbola.
37. If t1 and t2 are the roots of the equation x2 – 4x + 2 = 0, then the point of intersection
of tangents at ‘ t1 ‘ and ‘ t2 ‘ on xy = c2 is
(1) ,2 2 c c
(2) 2, 2 c c
(3) , 2 c c (4) , 2 c c
38. If e 1 and e 2 are the eccentricities of the hyperbolas xy = 9 and x 2 – y 2 = 25, then (e1,e2) lie on a circle C1with centre origin then the (radius))2 of the director circle of C1 is
(1) 2 (2) 4 (3) 8 (4) 16
39. Let α , γ be the roots of the equation t 1x 2 – 4 x + 1 = 0 and β, δ be the roots of t 2x 2 – 6 x + 1 = 0 and α, β, γ δ are in HP. Then the point of intersection of normals at ‘t1 ’ and ‘ t2’, on xy = c2 is
(1) 327921 , 264264 cc
(3) 232913921 , 264264 cc
Q. (40 –42)
(2) 237291 , 264264 cc
(4) none of these
A conic passes through the point (2, 4) and is such that the segment of any of its tangents at any point contained between the coordinate axes is bisected at the point of tangency.
40. The foci of the conic are (1) ()() 22,0 and 22,0 (2) ()() 22,22 and 22,22 (3) (4, 4) and (–4, –4) (4) ()() 42,42 and 42,42
41. The equations of directrices are (1) x + y = ±8 (2) x + y = ±4 (3) 42 xy+=± (4) x + y = ±1
42. The eccentricity of the conic is (1) 2 (2) 2 (3) 3 (4) 3 2
Q. (43 –45)
A point P moves such that sum of the slopes of the normal drawn from it to the hyperbola xy = 16 is equal to the sum of ordinates of feet of normal. The locus of P is a curve C.
43. Area of the equilateral triangle, inscribed in the curve C , having one vertex as the vertex of curve C is
(1) 7723 sq.units (2) 7763 sq.units
(3) 7603 sq.units (4) 7683 sq.units
44. The equation of the curve C is
(1) x2 = 4y (2) x2 = 16y
(3) x2 = 12y (4) y2 = 8x
45. If the tangent to the curve C cuts the coordinate axes at A and B, then the locus of the middle point of AB is_____.
(1) x2 = 4y (2) x2 = 2y
(3) x2 + 2y = 0 (4) x2 + 4y = 0
Q. (46 –48)
The difference between the second degree curve and pair of asymptotes is constant. If second degree curve represented by a hyperbola S=0, then equation of its asymptotes is S+λ=0 where λ is constant. which will be a pair of straight lines, then we get λ. Then equation of asymptotes is A ≡ S+λ=0 and if equation of conjugate hyperbola of S represented by S 1, then A is the arithmetic mean of S and S1
46. The asymptotes of a hyperbola having centre at the point (1, 2) are parallel to the lines 2x+3y =0 and 3x + 2y = 0. If the hyperbola passes through the point (5, 3), then its equation is
(1) (2x + 3y – 3)(3x + 2y –5) = 256
(2) (2x + 3y – 7)(3x + 2y –8) = 156
(3) (2x + 3y – 5)(3x + 2y –3) = 252
(4) (2x + 3y – 8)(3x + 2y –7) = 154
47. Pair of asymptotes of the hyperbola xy –3y – 2x = 0 is
(1) xy – 3y – 2x + 2 =0
(2) xy – 3y – 2x + 4 =0
(3) xy – 3y – 2x + 6 =0
(4) xy – 3y – 2x + 12 =0
48. If the angle between the asymptotes of hyperbola 2 2 22 1 xy ab −= is 3 π then the eccentricity of conjugate hyperbola is (1) 2 (2) 2 (3) 2 3 (4) 4 3
Q. (49 –51)
In a hyperbola, portion of tangent intercepted between asymptotes is bisected at the point of the contact. Consider a hyperbola whose centre is at origin. A line x + y = 2 touches this hyperbola at P(1, 1) and intersects the asymptotes at A and B such that AB 62 = units.
49. Angle subtended by AB at centre of the hyperbola is (1) 1 4 sin 5 (2) 1 2 sin 5 (3) 1 sin 5 3 (4) 1 sin 5 1
51. Equation of the tangent to the hyperbola at 7 1, 2
is
(1) 5x + 2y = 2
(2) 3x + 2y = 4
(3) 3x + 4y = 11 (4) 3x + 5y = 7
Matrix Match Questions
52. Let the circle ( x –1) 2 +( y –2) 2 =25 cuts a rectangular hyperbola with transverse axis along y = x at four points, A, B, C and D having coordinates ( x i, y i), i = 1, 2, 3, 4 respectively. O being the centre of the hyperbola. Now match the entries from the following two lists and choose the correct answer from the options given below:
List – I
List – II
(A) x1+x2+x3 +x4 is equal to (p) 2
(B) x12+x22+x3 2+x4 2 is equal to (q) 4
(C) OA2+OB2+OC2+OD2 is equal (r) 44
(D) y12+y22+y3 2+y4 2 is equal (s) 56 (t) 100
(A) (B) (C) (D)
(1) p r t s
(2) r p r t
(3) r r p t
(4) r r t p
53. If the normals at (xi, yi)(i = 1, 2, 3, 4) of the hyperbola xy = c2 meet at the point (p, q), then match the entries from the following two lists and choose the correct answer from the options given below
List – I
List – II
(A) p (p) x12+x22+x3 2+x4 2
(B) q (q) y12+y22+y3 2+y4 2
(C) p2 (r) x1+x2+x3 +x4
(D) q2 (s) y1+y2+y3 +y4
(A) (B) (C) (D)
(1) s s p p
(2) r s p q
(3) s p s q
(4) p s q s
54. Match the items of List-I with the items of List – II and choose the correct option.
List – I (Equation of Hyperbola)
List - II (Foci, eccentric)
(A) 9x2–16y2 = 36 (p) (±4, 0), 2
(B) 3x2 –y2 = 12 (q) (±5, 0), 5 3
(C) 16x2–9y2 = 144 (r) (0, ±3), 3 2
(D) 4x2–5y2+20 = 0 (s) 55 ,0, 24 ±
A B C D
(1) r, q, p, s
(2) s, p, q, r
(3) p, s, r, q
(4) s, s, p, q
55. If e1 and e2 are the roots of the equation x2 -a x+2 = 0, then match the entries from the following two lists and choose the correct answer from the options given below
List - I
List - II
(A) If e1 and e2 are the eccentricities of the ellipse and hyperbola respectively, then the value(s) of a is/are (p) 6
(B) If the both e1 and e2 are the eccentricities of the hyperbolas, then the value(s) of a is/are (q) 5 2
(C) If e1 and e2 are the eccentricities of hyperbola and conjugate hyperbola, then the value(s) of a is/are (r) 22
(D) If e1 the eccentricities of the hyperbola for which there exists infinite points from which perpendicular tangents can be drawn and e2 is the eccentricity of the hyperbola in which no such points exist, then the values of a are (s) 5
A B C D
(1) qr, ps, rp, rs
(2) ps, qr, r, ps
(3) rs, sp, pr, qs
(4) r, pq, p, ps
56. Match the items of List–I with the items of List – II and choose the correct answer.
List – I
(A) A variable tangents
to 2 2 1 4 x y −= cuts
2 2 1 4 x y += in points P &
List – II
Q. Locus of mid-points of PQ is (x2+4y2)2 = K(x2–4 y2)2, then K = (p) 1
(B) From a point (0, 2) a pair of tangents are drawn to x2 –y2 = 1 whose magnitude of the product of slopes (q) 4
(C) A circle of radius 2 unit is drawn through the points in which 2 2 1 32 xy −= &
2 2 2 1 x y a += intersect. Then a2 = (r) 5
(D) If a variable line has intercepts on the X & Y axes e&e' respectively; where e &e' are the eccentricities of a hyperbola and its conjugate hyperbola, then this line always touches the circle x2+y2 = r2, where r = (s) 6
A B C D
(1) q r p s
(2) q r s p
(3) r q p s
(4) r q s p
57. Match the items of List-I with items of the List-II choose the correct options List - I List - II
(A) A tangent drawn to hyperbola 2 2 22 1 xy ab −= at 6 P
forms a triangle of area 3a2 square units, with coordinate axes, then the square of its eccentricity is equal to (p) 17
(B) If the eccentricity of the hyperbola x2 –y2sec2θ = 5 is 3 times the eccentricity of the ellipse x2sec2θ+y2 = 25, then 4cos2θ is (q) 8
(C) For the hyperbola 2 2 3 3 x y −= , acute angle between its asymptotes is 24 L π , then value of 'L' is
(D) For the hyperbola xy = 8 any tangent of it at P meets co-ordinate axes at Q and R then area of triangle CQR where 'C' is centre of the hyperbola is
(r) 16
(s) 2
(t) 5
A B C D
(1) p s q r
(2) q p s r
(3) s q p t
(4) q s t r
58. Match the items of List-I with items of the List-II choose the correct options. List - I List - II
(A) The length of the latus rectum of the hyperbola 16 x2–9y2 = 144 is
(p) 2 3
(B) The product of the perpendiculars drawn from any point on the hyperbola x2–2y2 = 2 to its asymptotes is (q) 3
(C) The length of the transverse axis of the hyperbola xy = 18 is (r) 32 3
(D) The product of the lengths of the perpendiculars drawn from the foci of 3x2–4y2 = 12 on any of its tangents is
A B C D
(1) r p s q
(2) p q r s
(3) p q s t
(4) q r t p
(s) 12 (t) 5
59. The correct match from List-I to List-II in the following is List - I List - II
(A) For an ellipse 2 2 1 94 xy+= with vertices A and A', tangent drawn at the point P in the first quadrant meets the y-axis in Q and the chord A'P meets the y-axis in M. if O is the origin then OQ2 –MQ2 equals to
(p) 3
(B) If the product of the perpendicular distances from any point on the hyperbola 2 2 22 1 xy ab −= of eccentricity e = 3 from its asymptotes is equal to 6, then the length of the transverse axis of the hyperbola is (q) 4
(C) If F1 and F2 are the feet of the perpendiculars from the foci S1 and S2 of an ellipse 2 2 1 53 xy+= on the tangent at any point P on the ellipse, then(S1F1).
(S2F2) is equal to
A B C
(1) q r p
(2) p r q
(3) q p r
(4) r q p
(r) 6
BRAIN TEASERS
1. If two distinct tangents can be drawn from the point (α, 2)on different branches of the hyperbola 2 2 1 916 xy −= , then (1) 3 2 α< (2) 3 2 α> (3) | α | > 3 (4) α = 1
2. Let A (sec θ , 2 tan θ ) and B(sec φ , 2 tan φ ), where θ + φ = 2 π , be two points on the hyperbola 2 x 2 – y 2 = 2. If ( α , b ) is the point of the intersection of the normals to the hyperbola at A and B, then (2 b )2 is equal to ______
3. A point on rectangular hyperbola xy = k2 (k is purely imaginary) is ( a, α ) where a is a positive number, α can taken value (1) 1 (2) 3 2 (3) 17 (4) 12
4. The sides AC and AB of the D ABC touch the conjugate hyperbola of the hyperbola 2 2 22 1 xy ab −= . If the vertex A lies on the ellipse 2 2 22 1 xy ab += , then the side BC must touch (1) parabola (2) circle (3) hyperbola (4) ellipse
5. Let any point P, on the lower branch of the hyperbola satisfies PS–PS’ = 4, line passing through L(–2, 0) and S’ intersects the line passing through M(4, 0) and S in R, then the locus of R is 2x2–4x+3xy = 2λ, then λ = ________
6. Let hyperbola 2 2 :1, 13 Cxy−= , with the left and right focal points being F1 and F2 respectively. A line ℓ is drawn through F2
to intersect the right half of the hyperbola at points P and Q making 1 FPQ = 90°, then the inscribed circle radius of DF1PQ is (1) 32 (2) 5 –1
(3) 7 –1 (4) 73
7. P ( a sec θ , b tan θ ) and Q ( a sec φ , b tan φ ) are two points on the hyperbola 2 2 22 1 xy ab −= where φ + θ = 2 π . If (h, k) is the point of intersection of the normals drawn at P and Q, then k =
(1) 22ab b (2) 22ab b +
(3) 22ab b
(4)
22ab b
8. If a ray of light incident along the line 3 x–y(4 2 +5)+15 = 0 gets reflected from the hyperbola 2 2 1 169 xy −= , then its reflected ray goes along the line
(1) x 2 –y+5 = 0
(2) 2 y–x+5 = 0
(3) 2 y–x–5 = 0
(4) 3x+(5–4 2 )y = 15
9. If two tangents can be drawn to the different branches of Hyperbola x2–9y2 = 9 from the point ( α , b ) then ( α , b ) can be
(1) 1 2, 4
(3) 1 1, 2
(2) 1 1, 4
(4) (5, –1)
FLASHBACK (P revious JEE Q uestions )
JEE Main
1. Let e1 be the eccentricity of the hyperbola 22 1 169 xy −= and e 2 be the eccentricity of the ellipse 22 22 1, xy ab += a > b which passes through the foci of the hyperbola. If e1e2 = 1. then the length of the chord of the ellipse parallel to the x-axis and passing through (0, 2) is (27th Jan 2024 Shift 2)
(1) 45 (2) 85 3
(3) 105 3 (4) 35
2. Let P be a point on the hyperbola 22 :1, 94 xy H −= in the first quadrant such that the area of triangle formed by P and the two foci of H is 213. Then, the square of the distance of P from the origin is (30th Jan 2024 Shift 2) (1) 18 (2) 26 (3) 22 (4) 20
3. If the foci of a hyperbola are same as that of the ellipse 22 1 925 xy+= and the eccentricity of the hyperbola is 15 8 times the eccentricity of the ellipse, then the smaller focal distance of the point 142 2, 35
on the hyperbola, is equal to (31st Jan 2024 Shift 1)
(1) 28 7 53 (2) 24 14 53
(3) 216 14 53 (4) 28 7 53 +
4. For 0, 2 θπ<< If the eccentricity of the hyperbola x 2 – y 2cosec 2 θ = 5 is 7 times eccentricity of the ellipse x 2cosec 2 θ + y 2 = 5, then, the value of θ is (1st Feb 2024 Shift 1) (1)
5. The point () 26, 3 P lies on the hyperbola 2 2 22 1 xy ab −= having eccentricity
5 2 . If the tangent and normal at P to the hyperbola intersect its conjugate axis at the point Q and R respectively, then QR is equal to (26 th Aug 2021 Shift 2)
(1) 43 (2) 6 (3) 63 (4) 36
6. The locus of the mid–points of the chords of the hyperbola x2 – y2 = 4 , which touch the parabola y2 = 8x, is (26th Aug 2021 Shift 2)
7. Let λx – 2y = μ be a tangent to the hyperbola a2x2 – y2 = b2. Then 22 ab λµ
is equal to (24th Jun 2022 Shift 1)
(1) –2 (2) –4 (3) 2 (4) 4
8. Let the foci of the ellipse 2 2 1 167 xy+= and the hyperbola 2 2 1 14425 xy α −= coincide. Then the length of the latus rectum of the hyperbola is (25th Jul 2022 Shift 2) (1) 32 9 (2) 18 5 (3) 27 4 (4) 27 10
9. If the line x – 1 = 0, is a directrix of the hyperbola kx2 – y2 = 6, then the hyperbola passes through the point
(26th Jul 2022 Shift 2)
(1) () 25, 6 (2) () 5, 3
(3) () 5, 2 (4) () 25, 36
10. Let the tangent drawn to the parabola y2 = 24x at the point (α, β) is perpendicular to the line 2 x + 2 y = 5. Then the normal to the hyperbola 2 2 22 1 xy −= αβ at the point ( α + 4, β + 4) does not pass through the point
(26th Jul 2022 Shift 1)
(1) (25, 10) (2) (20, 12)
(3) (30, 8) (4) (15, 13)
11. The normal to the hyperbola 2 2 2 1 9 xy a −= at the point () 8,33 on it passes through the point (26th Jun 2022 Shift 2)
(1) () 15,23 (2) () 9,23
(3) () 1,93 (4) () 1,63
12. Let a tangent to the curve y2 = 24x meet the curve xy = 2 at the points A and B. Then the mid points of such line segments AB lie on a parabola with the (24th Jan 2023 Shift 1)
(1) length of latus rectum 3 2
(2) length of latus rectum 2
(3) directrix 4x = 3
(4) directrix 4x = –3
13. Let H be the hyperbola, whose foci are () 12,0 ± and eccentricity is 2 , then the length of its latus rectum is (31st Jan 2023 Shift 2)
(1) 3 (2) 5 1 (3) 2 (4) 3 2
14. Let P(x0, y0) be the point on the hyperbola 3x2 – 4y2 = 36 , which is nearest to the line 3x + 2y = 1, then () 00 2 yx is equal to (1 st Feb 2023 Shift 1)
(1) 3 (2) –3 (3) 9 (4) –9
15. Let the focal chord of the parabola P : y2 = 4x along the line L : y = mx + c, m > 0 meet the parabola at the points M and N . Let the line L be a tangent to the hyperbola H : x2 – y2 = 4 . If O is the vertex of P and F is the focus of H on the positive x–axis, then the area of the quadrilateral OMFN is
(29th Jul 2022 Shift 1)
(1) 26 (2) 214
(3) 46 (4) 414
16. Let the hyperbola H : 2 2 22 1 xy ab −= pass through the point () 22, 22 . A parabola is drawn whose focus is same as the focus of H with positive abscissa and the directrix of the parabola passes through the other focus of H. If the length of the latus rectum of the parabola is e times the length of the latus rectum of H, where e is the eccentricity of H, then which of the following points lies on the parabola?
(28th Jul 2022 Shift 2)
(1) () 23,32 (2) () 33,62
(3) () 3,6 (4) () 36,62
17. Let a > 0, b > 0. Let e and l respectively be the eccentricity and length of the latus rectum of the hyperbola 2 2 22 1 xy ab −= . Let e’ and l’ respectively be the eccentricity and length of the latus rectum of its conjugate hyperbola.
If ()2 2 1111 and ' 14 8 elel == ′ then the value of
77a + 44b is equal to (28th Jun 2022 Shift 2)
(1) 100 (2) 110 (3) 120 (4) 130
18. Let the eccentricity of the hyperbola
2 2 22 :1Hxy ab −= be 5 2 and length o f its latusrectum be 62 If y = 2 x + c is a tangent to the hyperbola H, then the value of c2 is equal to (28th Jun 2022 Shift 1) (1) 18 (2) 20 (3) 24 (4) 32
19. The locus of the midpoints of the chord of the circle, x2 + y2 = 25 which is tangent to the hyperbola, 2 2 1 916 xy −= is (16th Mar 2021 Shift 1)
(1) (x2 + y2)2 – 16x2 + 9y2 = 0
(2) (x2 + y2)2 – 9x2 + 144y2 = 0
(3) (x2 + y2)2 – 9x2 – 16y2 = 0
(4) (x2 + y2)2 – 9x2 + 16y2 = 0
20. A hyperbola passes through the foci of the ellipse 2 2 1 2516 xy+= and its transverse and conjugate axes coincide wi th major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is (25th Feb 2021 Shift 2)
(1) 2 2 1 916 xy −=
(2) 2 2 1 94 xy −=
(3) x2 –y2 = 9
(4) 2 2 1 925 xy −=
21. Let the latusrectum of the hyperbola 22 2 1 9 xy b −= subtend an angle of 3 π at the centre of the hyperbola. If b 2 is equa l to () 1, l n m + where l and m are co-prime numbers then l2+m2+n2 is equal to ______ (30 th Jan 2024 Shift 1)
22. Let the foci and length of the latus rectum of an ellipse 22 22 1, xy ab ab +=> be () 5,0± and 50 respectively. Then, the square of the eccentricity of the hyperbola 22 222 1 xy bab −= equals _____. (31st Jan 2024 Shift 1)
23. Let the hyperbola 2 2 2 :1 x Hy a −= and the ellipse E : 3x2+4y2=12 be such that the le ngth of the latus rectum of H is equal to the length of the latus rectum of E . If eH and eE are the eccentricities of H and E respectively, then the value of 12( eH2+eE2) is equal to (24 th Jun 2022 Shift 2)
24. Let the equation of two diameters of a circle x2 + y2 –2x + 2fy + 1 = 0 be 2px – y = 1 and 2x + py = 4p. Then the slope m ∈ (0, ∞ ) of the tangent to the hyperbola 3 x 2 – y 2 = 3 passing through the centre of the circle is equal to ______. (25th Jul 2022 Shift 1)
25. Let the eccentricity of the hyperbola 2 2 22 1 xy ab −= be 5 4 . If the equation of the normal at the point 812 , 5 5
on the hyperbola is 85, xyβλ+= , then λ–β is equal to (25th Jun 2022 Shift 2)
26. Let a line L 1 be tangent to the hyperbola 2 2 1 164 xy −= and let L2 be the line passing through the orig in a nd perpendicular to L1. If the locus of the point of intersection of L1 and L2 is (x2+y2 )2=αx2+βy2 then α+β is equal to____. (26th Jun 2022 Shift 2)
27. A common tangent T to the curves 2 2 1 :1 49 Cxy+= and 2 2 2 :1 42143 Cxy−= does not pass through the fourth quadrant. If T touches C1 at (x1,y1) and C2 at (x2, y2), then |2x1+x2| is equal to ______. (27th Jul 2022 Shift 2)
28. Let 2 2 :1 13 n Hxy nn −= ++ , n ∈ N . Let k be the smallest even value of n such that the eccentricity of Hk is a rational number. If I is the length of the latus rectum of H k, then 21l is equal to _____.
(11th Apr 2023 Shift 1)
29. Let m1 and m2 be the slopes of the tangents drawn from the point P (4, 1) to the hyperbola 2 2 :1 2516 Hyx−= . If Q is the point from which the tangents drawn to H have slopes | m 1| and | m 2| and they make positive intercepts α and β on the x–axis, then ()2 PQ αβ is equal to ____.
(13th Apr 2023 Shift 1)
30. The foci of hyperbola are(±2, 0) and its eccentricity is 3 2 . A tangent, perpendicular to the line 2x + 3y = 6, is drawn at a point in the first quadrant on the hyperbola. If the intercepts made by the tangent on the x- and y–axes are a and b respectively, then |6a| + |5b| is equal to______.
(13th Apr Shift 2023 2)
31. The vertices of a hyperbola H are (±6, 0) and its eccentricity is 5 2 . Let N be the normal to H at a point in the first quadrant and parallel to the line 222 xy+= . If ‘ d’ is t he length of the line segm ent of N between H and the y– axis then ‘d2’ is equal to (25th Jan 2023 Shift 1)
32. An ellipse 2 2 22 :1Exy ab += passes through the vertices of the hyperbola 2 2 : 1 4964 Hxy−=−
Let the major and minor axes of the ellipse E coincide with the transverse and conjugate axes of the hyperbola H, respectively. Let the product of the eccentricities of E and H be 1 2 . If k is the length of the latus rectum of the ellipse E, then the value of 113 k is equal to ______. (27th Jul 2022 Shift 1)
33. Let 2 2 22 :1Hxy ab −= , a > 0, b > 0 , be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is () 42214 + . If the ecentricity of H is 11 2 , then the value of a2 + b2 is equal to _____. (29th Jun 2022 Shift 1)
34. A square ABCD has all its vertices on the curve x2y2 = 1. The midpoints of its sides also lie on the same curve. Then, the square of area of ABCD is _____.
(18 th Mar 2021 Shift 1)
35. The locus of the point of intersection of the lines ()3430 kxky+−= and () 3430 xyk−−= is a conic, whose eccentricity is ___. (25th Feb 2021 Shift 1)
36. Let the eccentricity of an ellipse 2 2 22 1 xy ab += is reciprocal to that of the hyperbola 2x2–2y2 = 1. If the ellipse intersects the hyperbola at right angles, then square of length of the latusrectum of the ellipse is _____.
CHAPTER TEST – JEE MAIN
Section – A
1. For a hyperbola whose centre is at (1, 2) and asymptotes are parallel to lines 2x + 3y = 0, x + 2 y = 1, then equation of hyperbola passing through (2, 4) is
(1) (2x + 3y – 5)(x + 2y – 8) = 40
(2) (2x + 3y – 8)(x + 2y – 5) = 40
(3) (2x + 3y – 8)(x + 2y – 5) = 30
(4) (2x + 3y – 5)(x – 2y – 8) = 30
2. The centre of a rectangular hyperbola lies on the line y = 2x. If one of the asymptotes is x + y + c = 0, then the other asymptote is
(1) x – y – 3c = 0
(2) 2x – y + c = 0
(3) x – y – c = 0
(4) 3x – 3y – c = 0
3. The equation of the common tangent to the curves y2 = 8x and xy = –1 is
(1) 3y = 9x + 2
(2) y = 2x + 1
(3) 2y = x + 8
(4) y = x + 2
4. The focus of rectangular hyperbola (x – h)
(y – k) = p2 is
(1) (h – p, k – p)
(2) (h – p, k + p)
(3) (h + p, k – p)
(4) (h + 2 p, k + 2 p)
5. If chords of the hyperbola x2 – y2 = a2 touch the parabola y2 = 4ax, then the locus of the middle Points of these chords is the curve
(1) y2(x + a) = x3 (2) y2(x – a) = x3
(3) y2(x – 2a) = 3x3 (4) y2(x – 2a) = 2x3
6. Consider the set of hyperbola xy = k, k ∈ R.
Let e1 be the eccentricity when k = 4 and e2 be the Eccentricity when k = 9, then e1 – e2 is equal to
(1) –1 (2) 0 3) 2 (4) 3
7. If e is the eccentricity of the hyperbola 2 2 22 1 xy ab −= and θ is angle be tween the asymptotes, then cos 2
is equal to:
(1) 1 e e (2) 1 e e
(3) 1 e (4) 1 1 e +
8. If the circle x 2 + y 2 = a 2 intersects the hyperbola xy = c2 in four points P(x1, y1), Q(x2, y2), R(x3, y3), S(x4, y4), then which of the following is correct?
(1) x1 + x2 + x3 + x4 = 2c2
(2) y1 + y2 + y3 + y4 = 0
(3) x1 x2 x3 x4 = 2c4
(4) y1 y2 y3 y4 = 2c4
9. Let the major axis of a standard ellipse equals the transverse axis of a standard hyperbola and their Director circles have radius equal to 2R and R respectively. If e1 and e2 are the eccentricities of the Ellipse and hyperbola then the correct relation is (1) 22 12 46 ee−= (2) 22 1242ee−= (3) 22 21 46 ee−= (4) 22 12 24 ee−=
10. The equation to the chord joining two points (x1, y1) and (x2, y2) on the rectangular hyperbola xy = c2 is
11. Number of common tangent with finite slope to the curves xy = c2 and y2 = 4ax is:
(1) 0 (2) 1 (3) 2 (4) 4
12. Let F 1 , F 2 are the foci of the hyperbola
2 2 1 169 xy −= and F 3 , F 4 are the foci of its conjugate hyperbola. If e H and e C are their eccentricities respectively then the statement which holds true is:
(1) Their equations of the asymptotes are different
(2) eH > eC
(3) Area of the quadrilateral formed by their foci is 50sq. units.
(4) Their auxiliary circles will have the same equation
13. The locus of the foot of the perpendicular from the centre of the hyperbola xy = c2 on a variable tangent is:
(1) (x2 – y2)2 = 4c2xy (2) (x2 + y2)2 = 2c2xy
(3) (x2 – y2) = 4c2xy (4) (x2 + y2)2 = 4c2xy
14. An equilateral triangle PQR is inscribed in a rectangular hyperbola xy = 36, if its in centre lies on line y = 6 then circum centre is (1) (–6, 6) (2) (6, 6) (3) (2, 18) (4) (22, 3)
15. The ellipse 4x2 + 9y2 = 36 and the hyperbola a2x2 – y2 = 4 intersect at right angles. Then the equation of the circle through the points of intersection of two conics is (1) x2 + y2 = 5
(2) () 22 5340 xyxy+−−=
(3) () 22 5340 xyxy+++= (4) x2 + y2 = 25
16. Let ‘p’ be the perpendicular distance from the centre C of the hyperbola 2 2 22 1 xy ab −= to the tangent drawn at a point R on the hyperbola. If S and S’ are the two foci of the hyperbola, then () 2 2 12 2 1 b RSRSa p λ+=+
where l is
(1) 1 (2) 2 (3) 4 (4) 3
17. Let P(3, 3) be a point on the hyperbola, 2 2 22 1 xy ab −= . If the normal to it at P intersects the x–axis at (9,0) and e is its eccentricity, then the ordered pair ( a2 , e2) is equal to :
(1) 9 ,3 2
(3) 9 ,2 2
(2) 3 ,2 2
(4) (9, 3)
18. If the tangents drawn from a point on the hyperbola x2 – y2 = a2 – b2 to the ellipse 2 2 22 1 xy ab += make angles α and b with the transverse axis of the hyperbola, then (1) tanα – tan b = 1 (2) tanα + tan b = 1 (3) tanαtan b = 1 (4) tanαtan b = –1
19. If (λ – 2)x2+ 4y2 = 4 represents a rectangular hyperbola then λ = (1) 0 (2) 1 (3) –2 (4) 3
20. If the intercepts made by tangent, normal to a rectangular x2 – y2 = a2 with x–axis are a1, a2 and with y–axis are b1, b2 then a1a2 + b1b2 = (1) 0 (2) 1 (3) –1 (4) a2
Section – B
21. Let x2 + y2 = 4r2 and xy = 1 intersects at A and B in first quadrant if AB= 14 units then the value of |2r| is
22. e1, e2, e3, e4 are eccentricities of the conics xy = c2 , x2 – y2 = c2 , 2222 22221,1 yyxx abab −=−=− and 2222 1234 1111 sec eeee +++=θ , If 2θ = , k π then k =______.
23. A chord of the hyperbola x 2 – 2 y 2 = 1 is bisected at the point (–1, 1). If area of the triangle formed by the chord and the coordinate axes is Δ, then 16Δ is equal to ______.
24. If the product of perpendicular drawn from any point on the hyperbola x2 – 2y2 – 2 = 0 to its asymptotes is 3 K where K =_____.
CHAPTER TEST – JEE ADVANCED
2019 P 1Model
Section – A
[Multiple Option Correct MCQs]
1. e be the eccentricity of a hyperbola and f ( e ) be the eccentricity of its conjugate hyperbola and ( f ο f ο f ο ....of n times)( e ) is F(e). Then () 3 1 Fede ∫
(1) 4 if n is even (2) 4 if n is odd
(3) 22 if n is even (4) 22 if n is odd
2. If the circle x 2 + y 2 = a 2 intersects the hyperbola xy = c2 in four points P(x1, y1), Q (x2, y2), R (x3, y3), S (x4, y4), then
(1) x1 + x2 + x3 + x4 = 0
(2) y1 + y2 + y3 + y4 = 0
(3) x1 x2 x3 x4 = c4
(4) y1 y2 y3 y4 = c4
3. An ellipse intersects the hyperbola 2( x 2 –y 2 )= 1 orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the hyperbola are the coordinate axes, then
(1) the ellipse is x2 + 2y2 = 2
(2) foci of the ellipse are (±1, 0)
(3) the ellipse is x2 + 2y2 = 4
(4) foci of the ellipse are () 2,0 ±
4. If x, y ∈ R then the equation 3x4 – 2(19y + 8)x2 + (361y2 + 2(100 + y4) + 64) = 2(190y + 2y2) represents in rectangular Cartesian system.
(1) parabola
(2) hyperbola
(3) circle
(4) ellipse
25. If m is slope of the common tangents to the parabola y 2 = 24 x and the hyperbola 5x2 – y2=5 then |m|=___________
5. If the foci of hyperbola lies on the line y = x , one asymptote is y = 2 x and it is passing through the point (3, 4), then
(1) Equation of hyperbola is 2 x2 – xy + 2y2 = 38
(2) Equation of hyperbola is 2 x2 – 5xy + 2y2 + 10 = 0
(3) Eccentricity of hyperbola is 17 4
(4) Eccentricity of hyperbola is 10 3
6. A circle ‘C ‘ touch the transverse axis at a focus S of the hyperbola 2 2 1 43 xy −= , and pass through one end of the conjugate axis [i.e (0, b), b > 0] Another circle ‘D’ touch the transverse axis at a focus S’ of the hyperbola 2 2 1 43 xy −= , and pass through same end of the conjugate axis [i.e (0, b), b > 0] Both the circles C and D intersect at E and F . Then which of the following is /are true
(1) 4 3 EF =
(2) 2 3 EF =
(3) Circumradius of the triangle SES ’ is equal to 5 3
(4) Circumradius of the triangle SFS ’ is equal to 4 3
Section – B
[Numerical Value Questions]
7. C is the centre of the hyperbola
2 2 1 41 xy −= and ‘ A ‘ is any point on it. The tangents at A to the hyperbola meet the line x – 2y = 0 and x + 2y = 0 at Q and R respectively. The value of CQ.CR equals to _____.
8. Number of points on the hyperbola
2 2 1 2516 xy −= common to the circle x2 + y2 = 36 are
9. If distance between two parallel tangents having slope m drawn to the hyperbola
2 2 1 949 xy −= is 2, then the value of 8 m 2 is____
10. Let H:
2 2 22 1 xy ab −= , a > b > 0 be a hyperbola in xy –plane whose conjugate axis LM subtends an angle 60° at one of its vertices N . Let the area of triangle L MN be 43 then distance between foci of hyperbola H is________.
11. A normal to the hyperbola
2 2 1 41 xy −= has equal intercepts on positive x–axis and y –axis. If the normal touches the ellipse
2 2 22 1 xy ab += and if a2 + b2 is k, then 3k is equal to _____.
12. If y = mx + c is tangent to the hyperbola
2 2 22 1 xy ab −= having eccentricity 5, then the least positive integral value of m is
13. If equation of hyperbola whose conjugate axis is 5 and distance between its foci is 13, is ax 2 – by 2 = c where a and b are coprime natural numbers, then value of ab c is_____
14. The equation 2 2 1 94 xy λλ += represents a hyperbola when a < l < b, then ba ba +
where [.] denotes greatest integer function__
Section – C [Single Option Correct MCQs]
15. Locus of the feet of the perpendicular from centre of the hyperbola x2 – 4y2 = 4 upon a variable normal to it has equation ( x2 + y2)2 (4y2 – x2) = l x2y2 then l is (1) 24 (2) 25 (3) 26 (4) 23
16. If a ray of light incident along the lin e () 3 542 15 xy+−= gets reflected from the hyperbola 2 2 1 169 xy −= , then its reflected ray goes along the line
(1) 250xy−+=
(2) 250 yx−+=
(3) 250 yx−−=
(4) () 3425150 xy−++=
17. A variable straight line is such that the algebraic sum of perpendicular distances from the points of intersection of the ellipse 2x2 + y2 = 2 and hyperbola 2x2 – 4y2 = 1 is 0 . What is the sum of the coordinates of the fixed point through which the straight line always passes?
(1) 0 (2) 1 (3) 2 (4) 3
18. A tangent with slope m to the parabola y2 =4x which bisect exactly two distinct chords of hyperbola x2 – 4y2 – 4 = 0 drawn from (–2, 0), then which of the following cannot be true?
(1) 1 2 m = (2) 2 1 7 m
=−
(3) 1 2 m =− (4) m = 0
ANSWER KEY
JEE Main Level
Theory-based Questions
(16) 1 (17) 1 (18) 1 (19) 3
JEE Advanced Level
Brain Teasers
Flashback
Chapter Test – JEE Main
Chapter Test – JEE Advanced
INTRODUCTION TO 3D GEOMETRY CHAPTER 17
Chapter Outline
17.1 Coordinates of a Point in Space
17.2 Distance between Two Points in Space
17.3 Section Formula
17.4 Centres and Areas of Various Figures in 3D Geometry
17.5 Locus and Translation of Axes
Understanding 3D geometry is important because it helps us grasp how things relate in space, which matters in fields like architecture, engineering, and physics. It boosts problemsolving by letting us see shapes and structures more clearly, leading to new ideas and useful inventions in many different areas of work.
17.1 COORDINATES OF A POINT IN SPACE
A fundamental mathematical framework used to represent points in a two-dimensional plane or three-dimensional space is the rectangular cartesian coordinate system. It is made up of two or three perpendicular axes that meet at a single origin, commonly labelled x,y and z. The horizontal direction is represented by the x-axis, the vertical direction by the y-axis , and the depth or height by the z-axis.
The intersection of these axes forms the origin (0,0,0). Points are defined by their distances along each axis, aiding in geometry, physics, and engineering. It provides a standardised way to describe the position of objects, facilitating calculations, analyses, and visualisations in three-dimensional space.
17.1.1 Rectangular Cartesian Coordinate System
Let ' XOX
be three mutually perpendicular lines (in space) intersecting at a point O called origin.
Lines ' XOX
, ' YOY
, and ' ZOZ
are called x-axis, y-axis, and z-axis, respectively.
Plane passing through ' XOX
, ' YOY
is called XY-plane or XOY-plane.
Plane passing through ' YOY
, ' ZOZ
is called YZ-plane and the plane passing through ' XOX
, ' ZOZ
is called ZX-plane.
XY, YZ, and ZX-planes are called coordinate planes and these planes are mutually perpendicular.
Above system of coordinate axes is called rectangular cartesian coordinate system.
Coordinates of a Point in Space
Let P be a point in the space and PM be the perpendicular from P to the XOY-plane.
Let MN be the perpendicualr from M to the x-axis. Here, MN and y-axis are parallel.
Here, ON, NM, MP are called the x-coordinate, y-coordinate, z-coordinate of P, respectively. If ON = x, NM = y, and MP = z, then (x,y,z) are called the coordinates of P.
The coordinates of origin are (0, 0, 0).
Let P = (Px, Py, Pz).
i. P lies on the x -axis ⇔ P y = 0 and P z = 0
ii. P lies on the y -axis ⇔ P x = 0 and P z = 0
iii. P lies on the z -axis ⇔ P x = 0 and P y = 0
iv. P lies on the XOY-plane ⇔ P z = 0
v. P lies on the YOZ-plane ⇔ P x = 0
vi. P lies on the ZOX-plane ⇔ P y = 0
vii. If any point lies on x-axis, then its y, z coordinates are equal to zero and the point is of the form () ,0,0 x
viii. If any point lies on y-axis, then its x, z coordinates are equal to zero and the point is of the form () 0,,0 y .
ix. If any point lies on z-axis, then its x, y coordinates are equal to zero and the point is of the form () 0,0, z
Octants:
The three coordinate planes divide the space into eight equal parts, called octants The octant formed by the edges ,, OXOYOZ is called the first octant. We write it as OXYZ.
The octant whose bounding edges are OX, OY', OZ' is denoted by OXY'Z'. In a similar manner , the remaining six octants can be formed.
The coordinates of a point in a specific octant are determined by the octant in which it is located. Based on the positive and negative signs, each octant has its unique set of coordinate values. In the first octant () ,, +++ , for example, all three coordinates () ,, xyz are positive. The x-coordinate in the second octant () ,, −++ would be negative, but the y and z-coordinates would be positive. Similarly, the remaining octants have various positive and negative value combinations for each coordinate. The coordinates of a point inside an octant are determined by its position and the octant into which it falls.
The following table shows the octants and the sign of coordinates in each octant.
1. In which octant does the point () 1,4,2 lie ?
Sol. The given point is () 1,4,2 .
Since the x -coordinate and z -coordinate are negative, and y-coordinate is positive, the point () 1,4,2 lies on the octant OXYZ′′
Try yourself:
1. In how many octants, z-coordinates will be positive?
Ans: 4
TEST YOURSELF
1. The point (2, –4, –7) lies in the octant (1) OXY'Z' (2) OX'Y'Z (2) OXY'Z (4) OXYZ'
2. Which axis does the point (0, 4, 0) lie on? (1) x-axis (2) y-axis (3) z-axis (4) Origin
3. The point (–1, 4, –2) lie in the octant (1) OX'YZ' (2) OXY'Z' (3) OX'Y'Z (4) OX'Y'Z'
4. In which plane the point (0, 2, 3) lies in? (1) xy-plane (2) yz-plane (3) xz-plane (4) none of these
Answer Key (1) 1 (2) 2 (3) 1 (4) 2
17.2 DISTANCE BETWEEN TWO POINTS IN
SPACE
The distance between two points
() 111 ,, Axyz and () 222 ,, Bxyz in space is ()()() 222 212121 ABxxyyzz =−+−+− .
Coordinates of a point which lies on x-axis and at a distance of a units from origin is () ,0,0a ± . Coordinates of a point which lies on y -axis and at a distance of b units from origin is () 0,,0 b± . Coordinates of a point which lies on z -axis and at a distance of c units from origin is () 0,0, c ± .
The distance of point () ,, Pxyz from
i. oigin is 222 xyz ++
ii. x-axis is 22 yz +
iii. y-axis is 22 xz +
iv. z-axis is 22 xy +
v. xy-plane is z
vi. yz-plane is x
vii. xz-plane is y
Applications of Distance Formula
Let ,, ABC be any three points. If the sum of any two distances of () ,, ABBCCA is equal to third distance, then those three points are collinear.
Figure formed by the given three points:
Let ,, ABC be any three points. Sum of any two distances of () ,, ABBCCA is more than the third distance, then those three points form a triangle. Use the properties of triangle to determine its name.
To identify the figure formed by the given four points which lie in the same plane, first find the lengths of sides and diagonals and then use the properties of quadrilaterals.
2. Find the value of x , if the distance between the points () 5,1,7 and () ,5,1 x is 9 units.
Sol. Given: () 5,1,7 and () ,5,1 x , and distance between them is 9 units.
Hence,
()()() 222 551179 x −+++−=
Squaring on both sides, () () 2 2 57281 59 53 x x x −+= ⇒−= ⇒−=±
Therefore, x = 8 or 2
Try yourself:
2. Show that the points A(–4, 9, 6), B(–1, 6, 6), and C(0, 7, 10) form a right-angled isosceles triangle.
TEST YOURSELF
1. The distance between the point (9, –8, 15) from the x-axis is (1) 17 (2) 34 (3) 25 (4) 20
2. If the extremities of a diagonal of a square are (1, –2, 3), (2, –3, 5), then the length of its side is (1) 6 (2) 3 (3) 11 (4) 13
3. The point on y-axis which is equidistant to (2, 0, 6), (–6, 2, 4) is (1) (1, 2, 0) (2) (0, 4, 0) (3) (0, –4, 0) (4) (–1, 2, 0)
4. The points P(0, 7, 10), Q(–1, 6, 6), C(–4, 9, 6) are the vertices of an (1) isosceles right-angled triangle (2) obtuse triangle (3) equilateral triangle (4) none
5. The points A(3, 2, –4), B(5, 4, –6), and C(9, 8, –10) form (1) an isosceles triangle (2) an equilateral triangle (3) collinear points (4) a right angled triangle
6. The points A(2, –1, 4), B(1, 0, –1), C(1,2,3) and D(2, 1, 8) form a (1) rectangle (2) square (3) rhombus (4) parallelogram
7. The points A(5, –1, 1), B(7, –4, 7), C(1, –6, 10), D(–1, –3, 4) form (1) a parallelogram (2) a rhombus (3) a square (4) a rectangle
Answer Key
(1) 1 (2) 2 (3) 2 (4) 1 (5) 3 (6) 4 (7) 3
17.3 SECTION FORMULA
The coordinates of a point which divides the line segment joining the points () 111 ,, Axyz and () 222 ,, Bxyz in the ratio : mn internally
The coordinates of a point which divides the line segment joining the points () 111 ,, Axyz and () 222 ,, Bxyz in the ratio : mn externally is 212121 ,, mxnxmynymznz mnmnmn where m–n ≠ 0
17.3.1 Midpoint and Trisection Points of a Line Segment
Let () 111 ,, Axyz and () 222 ,, Bxyz be two points.
1. The midpoint of the line segment AB is 121212 ,, 222 Cxxyyzz +++
2. One of the trisecting points of the line segment AB is 212121 222 ,, 333
Pxxyyzz +++ ; this is nearer to A. Another trisecting point is 121212 222 ,, 333
Qxxyyzz +++ ; this is nearer to B.
17.3.2 Ratios that Coordinate Axes or Coordinate Planes Dividing the Line Segment
Let () 111 ,, Axyz and () 222 ,, Bxyz be two points.
1. x -axis divides the line segment AB in the ratio 12 : yy or 12 : zz . If these two ratios are not equal, then x-axis does not intersect the line AB
2. y -axis divides the line segment AB in the ratio 12 : xx or 12 : zz . If these two ratios are not equal, then y-axis does not intersect the line AB .
3. z -axis divides the line segment AB in the ratio 12 : xx or 12 : yy , if these two ratios are not equal, then z-axis does not intersect the line AB .
4. XY plane divides the line segment AB in the ratio 12 : zz .
5. YZ plane divides the line segment AB in the ratio 12 : xx
6. XZ plane divides the line segment AB in the ratio 12 : yy
7. If () ,, Pxyz be any point on the line joining the points A(x1, y1, z1), B(x2, y2, z2) then P divides line AB in the ratio 12 : xxxx (or) 12 : yyyy (or) z1 –z : z–z2
17.3.3 Collinear Points
If the points A( x 1 , y 1 , z 1 ), B( x 2 , y 2 , z 2 ), and C(x3, y3, z3) are collinear points, then AB : BC = (x1 – x2) : (x2 – x3) or (y1 – y2) : (y2 – y3) or (z1 – z2) : (z2 – z3) or 121212 232323 xxyyzz xxyyzz ==
The condition that three points
()()() 111222333 ,,,,,,,, AxyzBxyzCxyz are
to be collinear is 111 222 333 0. xyz xyz xyz =
3. If the point (()()() ) 121121121 ,, xtxxytyyztzz +−+−+− divides the line segment joining the points () 111 ,, xyz and () 222 ,, xyz internally, then find the ratio.
Sol. The given point can be written as (()()()) 212111 1,1,1 txtxtytytztz +−+−+−
Hence, the ratio in which the given point divides the line segment joining the points () 111 ,, xyz and () 222 ,, xyz is 1:. tt
Try yourself:
3. If the line joining the points () 1,3,4 A and B is divided by the point () 2,3,5 in the ratio 1 : 3, then find B.
Ans: )( 11,3,8 B
TEST YOURSELF
1. The point dividing the join of (3, –2, 1) and (–2, 3, 11) in the ratio 2 : 3 is (1) (1, 1, 4) (2) (1, 0, 5) (3) (2, 3, 5) (4) (0, 6, –1)
2. The coordinates of the point which divides the line joining the points (2, 3, 4) and (3, –4, 7) in the ratio 2 : 4 externally is (1) (10, 1, 1) (2) (1, 10, 1) (3) (10, –10, 10) (4) (1, 1, 10)
3. If A = (1, 2, 3) and B = (3, 5, 7) and P,Q are the points on AB such that AP = PQ = QB, then the midpoint of PQ is (1) (2, 3, 5) (2) (2, 3.5, 5) (3) (2, 4, 5) (4) (4, 7, 10)
4. Points A (3, 2, 4), 332838 ,,, 555
B and C(9, 8, 10) are given. The ratio in which B divides AC is (1) 5 : 3 (2) 2 : 1 (3) 1 : 3 (4) 3 : 2
17: Introduction to 3D Geometry
5. The harmonic conjugate of (2, 3, 4) with respect to the points (3, –2, 2), (6, –17, –4), is
(1) 111 ,, 234
(3) 1854 ,, 545
(2)
184 ,5, 55
(4)
184 ,5, 55
6. Let A(1, –1, 3), B(2, 3, 5) be divided by P in the ratio 3 : 4 internally. Then, harmonic conjugate of P w.r.t. A, B is (1) (2, 13, 3)
(2) (2, –13, 3)
(3) (–2, 13, –3) (4) (–2, –13, –3)
7. If A =(–2, 3, 4), B = (1, 2, 3) are two points and P is the point of intersection of AB and zx-plane, then P x +P y +P z = _____.
8. The line passing through the points A(5, 1, a) and B(3, b, 1) crosses the yz-plane at the point 1713 0,,. 22
Then, the value of a + b is _____.
9. If the x -coordinate of P on the join of Q(2,2,1) and R(5,2,–2) is 1, then z-coordinate of P is _____.
The process of finding the centres of a triangle in 3D is the same as the process in 2D, with an extension into the third dimension.
The centroid of the triangle formed by the points ()() 111222 ,,,,, AxyzBxyz , and () 333 ,, Cxyz is
,,
If G is centroid of the triangle ABC , then 3G = A + B + C and 3 CGAB =−− .
Centroid divides the line OS in the ratio 2:1, where O is orthocentre and S is circumcentre.
If the internal angular bisector of angle A of triangle ABC intersects the opposite side BC in D and I is incentre of the triangle, then
i. BD : DC = AB : AC
ii. AI : ID = AB+AC : BC
17.4.1 Tetrahedron
Let ,, ABC be three non-collinear points in the space, and D be another point which is not in the plane of ,, ABC . So, the figure formed by the four points ,,, ABCD is called tetrahedron.
1. The tetrahedron ABCD has Four faces, namely ,,, ABCACDABDBCD ∆∆∆∆
2. It has six edges: ,,,,, ABACBCADBDCD
3. It has four vertices: ,,, ABCD
4. The centroid G of tetrahedron ABCD divides the line joining any vertex to the centroid of its opposite triangle in the ratio 3:1.
5. The centroid of the tetrahedron formed by the points A ( x 1, y 1, z 1), B ( x 2, y 2 , z 2 ), C ( x 3 , y 3 , z 3 ), and D ( x 4 , y 4 , z 4 ) is 123412341234 ,, 444 Gxxxxyyyyzzzz +++++++++ =
6. The centroid G of tetrahedron ABCD divides the line segment joining vertex A to the centroid G1 of its opposite triangle BCD in the ratio 3:1.
17.4.2 Area of Triangle
The area of the triangle formed by the points ,, ABC is 1 2 ABAC ×
. The area of the triangle is equal to three times the area of the triangle formed by two vertices and centroid.
Area of the triangle formed by origin O and the points A ( x 1 , y 1 , z 1 ), B ( x 2 , y 2 , z 2 ) is
3. If (2, 3, 4) is the centroid of the tetrahedron for which (2, 3, –1), (3, 0, –2), (–1, 4, 3) are three vertices, then the fouth vertex is (1) (4, 5, 16) (2) (3, 2, 4) (3) (2, 3, 4) (4) (2, 2, 12)
4. If (0, 0, 0,), ( 3, 0, 0), and (0, 4, 0) are the vertices of a triangle, its incentre is (1) (1, 0 ,0) (2) ( 1, 1, 0) (3) (1, 1, 1) (4) ( 0, 0, 1)
4. Find the distance between the orthocentre and circumcentre of the triangle formed by the points ()()() 1,2,3,2,3,1,3,1,2.
Sol. Since the given vertices form an equilateral triangle, and all centres in equilateral triangle are coincident, the distance between the circumcentre and orthocentre is zero.
Try yourself:
4. If ()()() 0,1,2,2,1,3,1,3,1ABC are the vertices of a triangle, then find the distance between the orthocentre and circumcentre.
Ans: 3 2 units
TEST YOURSELF
1. If (1,0,3), (2,1,5), (–2,3,6) are the midpoints of the sides of a triangle, then the centroid of the triangle is (1) 1414 ,, 333 (2) 1414 ,, 333
(3) 1414 ,, 333
(4) 1414 ,, 333
2. The point (2, –1, 3) is the centroid of the triangle formed by the points A(3, –5, 2), B(–7, 4, 5), and C. The coordinates of C are (1) (9, 2, –2) (2) (10, –2, 2) (3) (2, 1, –3) (4) (8, 1, 9)
5. The circumcentre of the triangle formed by the points (1, 2, 3), (3, 1, 2), (2, 3, 1) is (1) (2, 2, 2) (2) (3, 3, 3) (3) (1, 1, 1) (4) (0, 0, 0)
6. The orthocentre of a triangle formed by the points (2, 1, 5), (3, 2, 3), (4, 0, 4) is (1) (2, 1, 5) (2) (3, 2, 3) (3) (4, 0, 4) (4) (3, 1, 4)
7. The perimeter of the triangle formed by the points (1, 0, 0), (0, 1, 0),(0, 0, 1) is (1) 2 (2) 22 (3) 32 (4) 42
8. The area of the triangle whose vertices are (1, 1, 1), (1, 2, 3), (2, 0, 1) is (1) 1 4 (2) 3 2 (3) 3 2 (4) 3 2
9. If (2, 3, 5), (–1, 3, 2), (3, 5, –2) are three consecutive vertices of a parallelogram, then its area is (1) 32 (2) 182 (3) 202 (4) 92
10. If (2, k , –1) is the centroid of the triangle with vertices (2, –1, 2), (1, –3, –4) and (3, k, –1), then | k | = ______.
11. If the incentre of ∆ ABC with vertices A(0, 0, 4), B(3, 0, 4), C(0, 4, 4) is (p,q,r), then p+q+r = ______.
The set of all points in the space satisfying given condition or a given property is called locus.
If () ,, Pxyz is any point in a locus, then the algebraic relation between ,, xyz obtained by using geometrical condition is called the equation of the locus.
1. The locus of a point which is at a distance of k units from xy-plane is zk =
2. The locus of a point which is at a distance of k units from yz-plane is xk = .
3. The locus of a point which is at a distance of k units from xz-plane is yk =
4. The locus of a point which is equidistant from xy-plane and yz-plane is 22 0 xz−= .
5. The locus of a point which is equidistant from yz-plane and xz-plane is 22 0 xy−= .
6. The locus of a point which is equidistant from xz-plane and xy-plane is 22 0 yz−=
The transformation obtained by shifting origin to some other point without changing the direction of axes is called translation of axes. If the axes are translated to the point () ,, hkl without changing the direction of coordinate axes and the coordinates of a point () ,, xyz are transformed to () ,, XYZ , then the relation between old and new coordinates is ,, xXhyYkzZl =+=+=+ .
5. Find the coordinates of the point (3, –7, 5) in the new system when the origin is shifted to the point (–1, –1, –1).
Sol: Given: Origin is shifted to the point (–1, –1, –1) and (x, y, z)=(3, –7, 5).
New coordinates of the point are X=3+1=4, Y=–7+1=–6, Z=5+1=6
∴ (x, y, z) = (4, –6, 6)
Try yourself:
5. Find the equation of locus of the point which is moving at equal distance from the points () 1,3,4 and () 1,3,4 .
Ans: 0 y =
TEST YOURSELF
1. The locus of the point which is at a distance of 3 units from zx-plane is
(1) | x | = 3 (2) | z | = 3
(3) | y | = 3 (4) x + y = 3
2. The equation of locus of a point whose distance from the y- axis is equal to its distance from the point (2, 1, –1) is
(1) x2 + y2 + z2 = 6
(2) x2 – 4x + 2z + 6 = 0
(3) y2 – 2y –4x + 2z + 6 = 0
(4) x2 + y2 – z2 = 0
3. A point P moves so that the points A,B, C taken on X,Y,Z axes are such that PA, PB,PC are mutually perpendicular and the volume of the tetrahedron OABC is k, where O is the origin. The locus of P is
(1) (x2 + y2 + z2)3 = 48kxyz
(2) (x2 + y2 + z2)2 = 24kxyz
(3) (x2 + y2 + z2)4 = 6kxyz
(4) (x2 + y2 + z2) = 12kxyz
4. The locus of a point whose distance from x-axis is equal to the distance from the point
(1, –1, 2) is
(1) y2 + 2x – 2y – 2z + 6 = 0
(2) x2 + 2x – 2y – 2z + 6 = 0
(3) x2 – 2x + 2y – 4z + 6 = 0
(4) z2 – 2x + 2y – 4z + 6 = 0
5. If the ends of the hypotenuse of a rightangled triangle are (2, 0, –3), (0, 4, 1), then locus of third vertex is
(1) x2 + y2 + z2 – 2x + 4y + 2z – 3 = 0
(2) x2 + y2 + z2 – 2x – 4y + 2z – 3 = 0
(3) x2 – y2 – z2 – 2x – 4y + 2z – 3 = 0
(4) x2 + y2 + z2 + 2x – 4y + 2z – 3 = 0
6. The locus of the point P, such that PA2 + PB2 = 10, where A = (2, 3, 4), B = (–2, 3, 4) , is
(1) x2 + y2 + z2 – 6y – 8z + 24 = 0
(2) x2 + y2 + z2 – 5x + y – 6z + 24 = 0
(3) 2(x2 + y2 + z2) – x + y – 4z + 12 = 0
(4) x2 + y2 + z2 + x – y + 4z – 12 = 0
7. If the point (1, 2, 3) is changed to the point (2, 3, 1) through translation of axis, then the new origin is
8. The point to which the axes can be translated to eliminate first degree terms in the equation 2 x 2 – 2 y 2 + z 2 – 4 x + 8 y + 2z – 5 = 0 is (1) (1, 2, 1) (2) (1, 2, –1) (3) (–1, 2, 1) (4) (1, –2, 1)
9. The transformed equation of 2 x2 – 3y2 + z2 + 4x + 6y – 4z – 2 = 0, when the axes are translated to the point (–1, 1, 2), is
CHAPTER REVIEW
Coordinates of a Point in Space
1. There are three coordinate axes and three coordinate planes in three-dimensional space.
2. Any two of three axes combine to produce a coordinate plane, resulting in a flat surface. The coordinate planes are xy -plane, yz-plane, and xz-plane.
3. The coordinates of origin are (0, 0, 0)
Let P = (Px, Py, Pz). Then,
• P lies on the x -axis ⇔ P y = 0 and P z = 0
• P lies on the y -axis ⇔ P x = 0 and P z = 0
• P lies on the z -axis ⇔ P x = 0 and P y = 0
• P lies on the XOY-plane ⇔ P z = 0
• P lies on the YOZ-plane ⇔ P x = 0
(1) 2x2 – 3y2 + z2 = 10 (2) 2x2 – 3y2 + z2 = 5
(3) 2x2 – 3y2 + z2 = 15
(4) 2x2 + 3y2 + z2 = 5
10. If the locus of a point, which is equidistant from (2, 3, –1) and (–3, 4, –3), is ax + by + cz = d and a < 0, then a + b + c + d = _____.
11. ABC is a right angled triangle right-angled at A. If B(2, 0, 2) and C(0, –1, –1) are the two vertices and the equation to the locus of A is x2 + y2 + z2 + px + qy + rz = s, then p + q + r + s is equal to ____.
12. The locus of a point which is at a distance of 5 unit from xy-plane is z2 – k = 0. Then, k = ____.
• If any point lies on x-axis, then its y, z coordinates are equal to zero and the point is of the form () ,0,0 x .
• If any point lies on y-axis, then its x, z coordinates are equal to zero and the point is of the form () 0,,0 y
• If any point lies on z-axis, then its x, y coordinates are equal to zero and the point is of the form () 0,0, z .
• There are eight octants in threedimensional space. octants are areas of space that are divided into eight sections based on the positive and negative signs of the coordinates along each axis. Each octant represents a distinct set of positive and negative values along the x, y, and z axes.
17: Introduction to 3D Geometry
Distance between Two Points in Space
4. The distance between two points () 111 ,, Axyz and () 222 ,, Bxyz in space is ()()() 222 212121 ABxxyyzz =−+−+− .
5. The distance of point () ,, Pxyz from
• oigin is 222 xyz ++
•x-axis is 22 yz +
•y-axis is 22 xz +
•z-axis is 22 xy +
•xy-plane is z
•yz-plane is x
•xz-plane is y
6. Col linearity : Let ,, ABC be any three points. Sum of any two distances of () ,, ABBCCA is equal to third distance, then those three points are collinear.
7. The condition that three points ()()() 112233 ,,,,, AxyBxyCxy are to be collinear is 111 222 333 0 xyz xyz xyz =
8. Figure formed by the given three points; Let ,, ABC be any three points. Sum of any two distances of () ,, ABBCCA is more than third distance then those three points form a triangle. Use the properties of triangle to identify the figure.
9. To identify the figure formed by the given four points which lies in the same plane, first find the lengths of sides and diagonals and then use the properties of quadrilaterals.
Section Formula
10. The coordinates of a point which divides the line segment joining the points () 111 ,, Axyz and () 222 ,, Bxyz in the ratio : mn internally is 212121 ,, mxnxmynymznz mnmnmn +++ +++ .
11. The coordinates of a point which divides the line segment joining the points () 111 ,, Axyz and () 222 ,, Bxyz
in the ratio m:n externally is
212121 ,, mxnxmynymznz mnmnmn
, where m–n ≠ 0.
12. The midpoint of the line segment AB is 121212 ,, 222 Cxxyyzz +++
13. One of the trisecting points of the line segment AB is 212121 222 ,, 333 Pxxyyzz +++
, this is nearer to A
14. Another trisecting point is 121212 222 ,, 333 Qxxyyzz +++
, this is nearer to B
15. Let () 111 ,, Axyz and () 222 ,, Bxyz are two points.
•x-axis divides the line segment AB in the ratio 12 : yy or 12 : zz , if these two ratios are not equal, then x -axis does not intersect the line AB
•y-axis divides the line segment AB in the ratio 12 : xx or 12 : zz , if these two ratios are not equal, then y -axis does not intersect the line AB
•z-axis divides the line segment AB in the ratio 12 : xx or 12 : yy , if these
two ratios are not equal, then z -axis does not intersect the line AB
•xy-plane divides the line segment AB in the ratio 12 : zz
•yz-plane divides the line segment AB in the ratio 12 : yy
•xz-plane divides the line segment AB in the ratio 12 : zz
16. The ratio in which a point () ,, Pxyz divides the line joining the points A ( x 1, y1, z1), B(x2, y2, z2) in the ratio 12 : xxxx or 12 : yyyy (or) z1 –z : z–z2
17. If the points A(x1, y1, z1), B(x2, y2, z2), and C(x3, y3, z3) are collinear points, then AB : BC = (x1 – x2) : (x2 – x3) or (y1 – y2) : (y2 – y3) or (z1 – z2) : (z2 – z3) or 121212 232323 xxyyzz xxyyzz ==
Centers and Areas of Various Figures in 3D
Geometry
18. The centroid of the triangle formed by the points ()() 111222 ,,,,, AxyzBxyz and () 333 ,, Cxyz is 123123123 ,, 333 Gxxxyyyzzz ++++++
19. Let A,B,C are three non collinear points in the space, and D is another point which is not in the plane of A,B,C. So that the figure formed by the four points A,B,C,D is called tetrahedron.
• The tetrahedron A,B,C,D has Four faces namely ,,, ABCACDABDBCD ∆∆∆∆
• It has six edges: ,,,,, ABACBCADBDCD
• It has four vertices: ,,, ABCD
20. The centroid of the tetrahedron formed by the points A ( x 1 , y 1 , z 1 ),
B(x2, y2, z2), C(x3, y3, z3), and D(x4, y4, z4) is 123412341234 ,, 444 Gxxxxyyyyzzzz +++++++++ =
• The centroid G of tetrahedron ABCD divides the line segment joining the points 1G , centroid of triangle BCD and opposite vertex A in the ratio 3:1 .
21. The area of the triangle formed by the points A,B, is 1 2 ABAC ×
. The area of the triangle is equal to three times the area of the triangle formed by two vertices and centroid.
Locus and Translation of Axes
22. The locus of a point which is at a distance of k units from xy-plane is zk =
23. The locus of a point which is at a distance of k units from yz-plane is xk =
24. The locus of a point which is at a distance of k units from xz-plane is yk =
25. The locus of a point which is equidistant from xy-plane and yz-plane is 22 0 xz−=
26. The locus of a point which is equidistant from yz-plane and xz-plane is 22 0 xy−=
27. The locus of a point which is equidistant from xz-plane and xy-plane is 22 0 yz−=
28. The transformation that obtained by shifting origin to some another point with out changing the direction of axes is called Translation of axes.
29. If the axes are translated to () ,, hkl and the coordinates of a point () ,, xyz transformed to () ,, XYZ then the relation is ,, =+=+=+ xXhyYkzZl
Exercises
JEE MAIN LEVEL
Level – I
Distance between two Points
Single Option Correct MCQs
1. The distances from P(1, 2, 3) to coordinate axes are (1) 13,10,5 (2) 11,10,5 (3) 13,20,15 (4) 23,10,5
2. The point on x – axis which is equidistant to (2, –1, –4), (–4, 3, 0) is (1) 13 , 0, 0 3
(3) 1 , 0, 0 3
(2) 16 , 0, 0 3
(4) 26 ,1, 0 3
3. The points A(2, 9, 12), B(1, 8, 8), C(–2, 11, 8) and D(–1, 12, 12) form a (1) parallelogram (2) rhombus
(3) square (4) rectangle
4. The coplanar points (3, 2, 5), (2, 1, 1), (–1, 4, 1), (0, 5, 5) forms a (1) parallelogram (2) square (3) rhombus
(4) rectangle
5. The points (2, 1, 5), (3, 2, 3), (4, 0, 4) form (1) equilateral triangle (2) isosceles triangle (3) right angled triangle (4) right angled isosceles triangle
6. The points (–2, 3, 5), (1, 2, 3), (7, 0, –1) are (1) collinear (2) vertices of a right angled triangle (3) vertices of an equilateral triangle (4) vertices of an isosceles triangle
Section Formula
Single Option Correct MCQs
7. If the points A,B,C,D are collinear and C,D divide AB in the ratios 2 : 3, –2 : 3 respectively, then the ratio in which A divides CD is
(1) 5 : 1 (2) 2 : 3
(3) 3 :2 (4) 1 : 5
8. If the x -coordinate of P on the join of Q (2, 2, 1) and R (5, 2, –2) is 4, then z-coordinate of P is (1) –2 (2) –1 (3) 1 (4) 2
9. If A = (5, 4, 2), B = (6, 2, –1), C = (8, –2, –7), then the harmonic conjugate of A with respect to B and C is
(1) (7, 0, –3) (2) 135 ,1, 22
(3) 135 ,1, 22
(4) 111 ,3, 22
10. The 1st point of trisection of segment joining (3, –1, 2) and (9, 5, 2) is (1) (5, 1, 5) (2) (5, 1, 2) (3) (5, 1, 4) (4) (4, 1, 6)
11. The ratio in which (5, 4, –6) divides the line segment joining (3, 2, –4), (9, 8, –10) is
(1) 2 : 1 (2) 1 : 2
(3) 2 : 3 (4) 3 : 2
12. A = (2, 4, 5) and B = (3, 5, –4) are two points. If the xy-plane, yz-plane divide AB in the ratios a : b, p : q respectively then ap bq += (1) 23 12 (2) 7 12
(3) 7 12 (4) 22 15
13. If A = (a, 8, –2), B=(1, 4, 6) and C = (5, 2, 10) and If A divides BC in the ratio m:n then 2ma ___. n3 −=
(1) –7 (2) –2 (3) 1 (4) 3
14. The 2nd point of trisection of the segment joining (6, 2, –4) and (3, –1, –1) is (1) (4, 0, –2) (2) (4, 0, 2) (3) (4, 0, 0) (4) (4, 0, 1)
Numerical Value Questions
15. Let A (3, 2, –4) and B (9, 8, –10) be two points. Let P1 divide AB in the ratio 1 : 2 and P2 divide AB in the ratio 2 : 1. If the point P ( α , β, γ) divides P 1P 2 in the ratio 1 : 1, then α + 2β + 2γ = ______.
16. If the zx -plane divides the line segment joining (1, –1, 5) and (2, 3, 4) in the ratio p : 1, then [p + 1] = _____ here [.] indicates the greatest integer function
Centres and Areas of Various Figures in 3D
Geomentry
Single Option Correct MCQs
17. In a ∆ABC vertex A, mid point of BC are (1, 2, 3) and 55 ,,0 22 . Then centroid is (1) (1, –2, –1) (2) (2, 1, 1) (3) (1, –2, 1) (4) (2, –1, 1)
18. If the orthocentre and the centroid of a triangle are at (5, 2, –6) and (9, 6, –4) respectively then its circumcentre is (1) (11,8,–3) (2) (8,8,–3) (3) (11,8,3) (4) (11,–8,–3)
19. The circumradius of ∆ ABC with vertices A(2, –1, 1), B(1, –3, –5), C(3, –4, –4) is (1) 6 2 (2) 35 2 (3) 41 2 (4) 41
20. If the orthocenter and the centroid of a triangle are (–3, 5, 1), (3, 3, –1) then the circumcenter is (1) (6, 2, –2) (2) (0, 2, 0) (3) (6, –2, –2) (4) (–6, 2, 2)
21. If the orthocentre, circumcentre of a triangle are (–3, 5, 2), (6, 2, 5) respectively, then the centroid of the triangle is (1) (3, 3, 4) (2) 379 ,, 222
(3) (9, 9, 12) (4) 933 ,, 2,22
22. The orthocentre of the triangle formed by the points (2, –1, 1), (1, –3, –5), (3, –4, –4) is (1) (2, –1, 1) (2) (1, –3, –5) (3) (3, –4, –4) (4) 88 2,, ,33
23. The incentre of ∆ABC with vertices A(0, 0, 4), B(3, 0, 4), C(0, 4, 4) is (1) (1, 1, 1) (2) (1, 1, 2) (3) (1, 1, 3) (4) (1, 1, 4)
24. The distance between the orthocenter and circumcentre of the triangle formed by the points (1, 2, 3), (3, –1, 5) (4, 0, –3) (1)
(3)
25. If the centroid of the tetrahedron is 5 0,1, 2
and three vertices are (–2, 3, 4), (3, –4, 2), (2, –5, 1), then the fourth vertex is (1) (–3, 2, 3) (2) (1, 0, 5) (3) (0, –1, 5) (4) (5, –1, 0)
26. If H,G,S,I are respectively orthocenter, centroid, circumcenter and incentre of a triangle formed by the points (1, 2, 3), (2, 3, 1) and (3, 1, 2) then H+G+S+I = (1) (2, 2, 2) (2) (4, 4, 4) (3) (6, 6, 6) (4) (8, 8, 8)
17: Introduction to 3D Geometry
Locus and Translation of Axes
Single Option Correct MCQs
27. Locus of point for which the sum of squares of distances from the coordinate axes is 10 units
(1) x2 + y2 + z2 = 8
(2) x2 + y2 + z2 = 10
(3) x2 + y2 + z2 = 15
(4) x2 + y2 + z2 = 5
28. The locus of the point ( rsecα cosβ, rsecα sinβ, rtanα) is
(1) x2 + y2 – z2 = r2
(2) x2 + y2 + z2 = r2
(3) x2 + y2 – z2 = 2r2
(4) x2 – y2 + z2 = r2
29. The locus of the point P such that PA2 + PB2 = 2PC2 where A = (1, 3, 2), B = (2, 4,–3), C = (–2, 1, 3) is
(1) x2 + y2 + z2 – x + y – 4z + 12 = 0
(2) x2 + y2 + z2 – 5x + y – 6z + 29 = 0
(3) 2(x2 + y2 + z2) – x + y – 4z + 12 = 0
(4) 14x + 10y – 14z – 15 = 0
30. The locus of the point P such that PA 2 + PB2 = 10, when A = (2, 3, 4), B = (–2, 3, 4) is
(1) x2 + y2 + z2 – 6y – 8z + 24 = 0
(2) x2 + y2 + z2 – 5x + y – 6z + 24 = 0
(3) 2(x2 + y2 + z2) – x + y – 4z + 12 = 0
(4) x2 + y2 + z2 + x – y + 4z – 12 = 0
31. If P = (1, 2, –3), Q = (1, 0, –1) and translate the axes by Q is origin, then new coordinates of P is
32. The transformed equation of x 2 + y 2 + z 2 – 6x – 8y + 2z + 24 = 0 when the axes are translated to the point (3, 4, –1) is (1) 2x2 + 3y2 – z2 = 25
(2) x2 + y2 + z2 = 2
(3) 2x2 – 3y2 – z2 = 25 (4) x2 + y2 + z = 50
Numerical Value Questions
33. P is a variable point which moves such that PA : PB = 2 : 3. If A = (–2, 2, 3) and B = (13, –3, 13) and P satisfies the equation x2 + y2 + z2 + ux + vy + wz – 247 = 0, then u+v+w = _____.
34. If a point whose distance from y- axis is thrice its distance from the point ( 1, 2, –1) satisfies the equation 8x2 + 9y2 – 8z2 – 18x – 36y + 18z + k = 0, then k = ____.
35. The locus of the point which is at a distance of 5 unit from (2, 1, –3) is x2 + y2 + z2 – 4x + 2y + 6z – k = 0. Then k = ____.
Level – II
Single Option Correct MCQs
1. If A (1, 2, 3), B (2, -3, 1), C (3, 2, –1) are three vertices of a tetrahedron ABCD and 539 ,, 224 G
is its centroid then the point which divides GD in the ratio 1:2 is (1) (6, 1, 3) (2) 8 3,,3 3
(3) 12 ,,1 33
(4) 87 3,, 32
2. The ends of hypotenuse of a right angled triangle are (2, 0, –3), (0, 4, 1) then locus of third vertex is (1) 222 24230xyzxyz++−++−=
(2) 222 24230xyzxyz++−−+−=
(3) 222 24230xyzxyz−−−−+−=
(4) 222 24230xyzxyz+++−+−=
3. The distance between the circumcentre and the orhtocentre of ∆ABC with vertices A(2, 1, 5), B(3, 2, 3), C(4, 0, 4) is k then k=_____
(1) 6 (2) 6 2 (3) 26 (4) 0
4. If the midpoints of the sides AB, BC and CA of a triangle are respectively D(1, 2, -3), E(3, 0, 1), F(–1, 1, –4) then the centroid of the ∆ ADF is
(1) (–1, 2, –5) (2) 52 3,, 33
(3) (1, 0, –3) (4) 5 1,,3 3
5. The orthocenter of triangle whose vertices are A(a, 0, 0), B(0, b, 0), and C(0, 0, c) is ,, kkk abc then k is equal to (where a,b,c
≠ 0 )
(1) 1 222 111 abc ++
(2) 1 111 abc ++
(3) 222 111 abc ++
(4) 111 abc ++
6. If (cosα, 0, sinα)(cosβ, sinβ, 0)(0, cosγ, sinγ) are the vertices of the triangle, then orthocenter of the triangle is (1) (0, 0, 0)
7. A=(0,2,3), B=(2,-1,5), c(3,0,-3) are vertices of ∆ABC. If a,b,c are HG, GS, SH then their increasing order is (where H is orthocenter, S is circumcentre, G is centroid) (1) a,b,c (2) c,b,a (3) b,a,c (4) b,c,a
9. In ∆ ABC, if A = (0, 0, 4) ; AB =4, BC = 3, CA = 5, I = (1, 0, 1) is the incentre and the internal bisector of ∠ A intersects BC at D(α, β, γ), then α + β + γ = _____.
(1) 3 (2) 4 (3) 4 3 (4) 3 4
10. If the centroid of the tetrahedron OABC where A,B,C are the points (a, 2, 3), (1, b, 2) and (2, 1, c) be (1, 2, 3) then the point (a,b,c) is at distance 5 λ from the origin then l must be equal to _____.
(1) 3 (2) 5 (3) 4 (4) 2
11. If α,β,γ be the roots of the equation 2 x3 + 3x2 – 12x + 3 = 0 and A(α,β,γ), B(β,γ,α), C(γ,α,β ) represent vertices of a triangle ABC then the centroid of the triangle lies upon the line
(1) x = y = z (2) x = 2y = 3z
(3) x = –2y = 3z (4) x = y = –2z
12. A(3, 2, 0), B(5, 3, 2), C(–9, 6, –3) are three points forming a triangle and the internal bisector of ADBAC ∠ meets the side BC at D, then D =
(1)
(3)
8. Let A(1, 2, 3), B(–1, 4, 6),C(0, –6, 4), D(1, 1,,1) be the vertices of a tetrahedron and G be the centroid of tetrahedron, G1 be the centroid of triangle BCD, then 1AG AG (1)
19577 ,, 81616
19717 ,, 81616
(2)
(4)
195717 ,, 81616
195717 ,, 866
13. If A(–1, 6, 6), B(–4, 9, 6), () 1 -5,22,22 3 G =
and G is the centroid of ∆ ABC, then the name of the triangle ABC is
(1) isosceles
(2) right –angled
(3) equilateral
(4) right angled isosceles
14. Match the following List-I List-II
(A) Centroid of the triangle with vertices
(p) (1,6,5)
A(2, 3, 7), B(6, 7, 5), C(1, 2, 3)
(B) Mid-point of the line joining the points
A(7, 9, 11) and B(–5, 3, –1) (q) (3,4,5)
(C) If A,B,C are projections of P(5,–2,6) on coordinate axes then centriod of
∆ ABC is (r) (3,3,2)
(D) Coordinates of the point dividing the join of (5,5,0) and (0,0,5) in the ratio 2:3
(s) 52 ,,2 33
(A) (B) (C) (D)
(1) q p s r
(2) q s p r
(3) s r q p
(4) q r s p
15. If the point [x1 + t(x2 – x1), y1 + t(y2 – y1), z 1 + t(z 2 – z 1 )] divides the line segment joining the points (x1, y1, z1) and (x2, y2, z2) Internally then
THEORY-BASED QUESTIONS
Very Short Answer Questions
1. In how many octants x -coordinate of a point is negative?
2. What is the formula for the distance of a point () ,, Pxyz from the origin?
3. What is the distance of a point () ,, Pabc from x-axis?
4. What is the formula for the area of a triangle formed by three points ,, ABC in three-
(1) t < 0 (2) 0 < t < 1
(3) t > 1 (4) t = 1
16. Let A(5, 4, 6), B(1, –1, 3) and C(4, 3, 2) form ∆ ABC. If the internal bisector of angle A meets BC in D, then the length of AD is
(1) 1 170 8 (2) 3 170 8
(3) 5 170 8 (4) 7 170 8
Numerical Value Questions
17. In ΔleABC mid points of the sides AB,BC, CA are respectively (α, 0, 0)(0, β, 0)(0, 0, γ). Then 222 222 ABBCCA αβγ ++ = ++ ________
18. In tetrahedron ABCD, A = (1, 2, –3) and B(–3, 4, 5) is the centroid of the tetrahedron. If P is the centroid of the ∆BCD then 3 21 AP =
19. A = (3, 4, 0), B = (5, 3, 2), C = (9, 6, –3) are the three points forming triangle then ratio in which internal bisector of BAC∠ divides BC is P : Q then P+Q =__________(P,Q are relatively prime).
20. Let A(3, 2, –4) and B(9, 8, –10) be two points. Let P1 divides AB in the ratio 1:2 and P2 divides AB in the ratio 2:1. If the point P( a,b,g ) divides PP12 in the ratio 1:1, then a+2b+2g = _______.
dimensional geometry?
5. What is the locus of a point which is moving at a distance of k units from origin?
Statement Type Questions
Each question has two statements: statement I (S-I)and statement II (S-II). Mark the correct answer as
(1) if both statement I and statement II are correct
(2) if both statement I and statement II are incorrect
(3) if statement I is correct, but statement II is incorrect
(4) if statement I is incorrect, but statement II is correct
6. S - I : If the x-coordinate of a point is zero, then that point lies on yz-plane.
S - II : If the x-coordinate of a point is zero, then that point may lies on either y-axis or on z-axis
7. S - I : One of the trisecting points of the line segment joining () 111 ,, Axyz and () 222 ,, Bxyz is 212121 222 ,, 333 Pxxyyzz +++
, this is nearer to A
S - II : One of the trisecting points of the line segment joining () 111 ,, Axyz and () 222 ,, Bxyz is 121212 222 ,, 333 Qxxyyzz +++
, this is nearer to B
8. S - I : The ratio in which yz-plane divides the line segment joining the points
A(x1, y1, z1), B(x2, y2, z2) is 12 : yy
S - II : The ratio in which xz-plane divides the line segment joining the points
A(x1, y1, z1), B(x2, y2, z2) is 12 : zz
9. S - I : The centroid of the tetrahedron divides the line segment joining any vertex to the centroid of its opposite triangle in the ratio 3 : 1.
S - II : The centroid of tetrahedron formed by A,B,C,D is 4 ABCD G +++ =
10. S - I : The locus of a point which is at a distance of k units from xy-plane is zk =
S - II : The locus of a point which is at a distance of k units from yz-plane is xk =
Assertion and Reason Questions
In each of the following questions, a statement of Assertion (A) is given, followed by a corresponding statement of Reason (R). Mark the correct answer as
(1) if both (A) and (R) are true and (R) is the correct explanation of (A)
(2) if both (A) and (R) are true but (R) is not the correct explanation of (A)
(3) if (A) is true but (R) is false
(4) if both (A) and (R) are false
11. (A) : The orthocentre of the triangle formed by the points ()()() 2,1,5,3,2,3,4,0,4 is () 3,1,4
(R) : The orthocentre of equilateral triangle coincide with the centroid of it.
12. (A) : The points ()()() 1,1,1,1,2,3,2,0,1 are collinear
(R) : The condition that three points ()()() 111222333 ,,,,,,,, AxyzBxyzCxyz are to be collinear is 111 222 333 0 xyz xyz xyz ≠
13. (A) : If () 2,1,3 , () 3,1,5 and () 1,2,4 are the mid points of the sides ,, BCCAAB of triangle ABC respectively, then the perimeter of the triangle is () 2563 ++ units.
(R) : If ,, DEF are midpoints of sides of a triangle ABC , then the perimeter of triangle ABC is double the perimeter of the triangle DEF
14. (A) : In an equilateral triangle centroid, circumcentre, and orthocentre coincide.
(R) : In right angled triangle, the vertex at right angle is orthocentre, midpoint of hypotenuse is circumcentre,
15. (A) : The locus of a point which is equidistant from xy-plane and xzplane is 22 0 xy−=
JEE ADVANCE LEVEL
Multiple Option Correct MCQs
1. A(2, –3, 4) and B(–5, 1, –7) then the ratio in which AB divides (1) xy-plane, 4 : 7 (2) zx-plane, –2 : 5 (3) xz-plane, 3 : 1 (4) xy-plane, –4 : 7
2. If H,G,S,I are orthocenter, centroid, circumcentre, and incentre of a triangle formed by points (1, 2, 3), (2, 3, 1), (3, 1, 2) then
(1) S = (2, 2, 2)
(2) H+I= (4, 4, 4)
(3) H+G+S+I= (8, 8, 8)
(4) H–G–S+I= (0, 0, 0)
3. If O(0, 0, 0) is the orthocenter of a triangle formed by the points (cosα, sinα, 0), (cosb, sin b , 0), (cos g , sin g , 0) then
(1) cosα + cos b + cos g=0
(2) sinα + sin b + sin g=0
(3) () Cos23 αβγ ∑−−=
(4) () Sin20 αβγ ∑−−=
Numerical/Integer Value Questions
4. If (1, 1, a ) is the centroid of the triangle formed by the points (1, 2, –3), ( b , 0, 1) and (–1, 1, –4), then b – a = ____.
5. The distance between the circumcentre and the orthocentre of ∆ABC with vertices A(2, 1, 5), B(3, 2, 3), C(4, 0,4) ____.
6. The ortho centre of triangle formed by (2, 1, 5), (3, 2, 3), (4, 0, 4) is ( α, β, γ) ⇒ α + β + γ = _____
(R) : The perpendicular distance of a point () ,, Pxyz from xy -plane is z and the distance of () ,, Pxyz from xz-plane is x
7. A(5, 4, 6), B(1, –1, 3), C(4, 3, 2) form ∆ABC. If the internal bisector of angle A meets BC in D. The length a ADcabc b =⇒++ = _____ (a,b are coprime).
8. If origin is the orthocenter of a triangle formed by the points (cosα, sinα, 0), (cosβ, sinβ, 0)(cosγ, sinγ, 0), then ∑cos(2α – β – γ) = ____.
Passage-based Questions
(Q. 9 – 11)
Let A(1, 1, 1), B(3, 7, 4) and C(–1, 3, 0) be three points in a plane.
9. Centroid of triangle ABC is (1) 115 1,, 33
(2) (1, –2, 5)
(3) (3, 6, 7)
(4) (3, 11, 5)
10. Length of the median passing through C is (1) 22 4 (2) 71 9 (3) 1 2 (70) 4 (4) 65 2
11. Area of the triangle ABC is (in sq. units)
(1) 23571 14 (2) 20384 14
(3) 25312 14 (4) 14321 14
(Q. 12 – 14)
Let A(–2, 2, 3) and B(13, –3, 13) be two points and L be a line passing through the point A.
12. Coordinates of point P which divides the join of A and B in the ratio 2 : 3 internally are
(1) 332 ,,9 55
(2) (4, 0 , 7)
(3) 321217 ,, 555
(4) (20, 0, 35)
13. The first point of trisection of line segment AB is
(1) (3, 1/3, 19/3)
(2) (11/2, –1/2, 8)
(3) (8, –4/3, 29/3)
(4) (8, 6, 9)
14. The locus of a point P( x , y , z ) such that PA=PB, is
(1) 3x+y–2z=13 (2) 3x–y+2z=33
(3) 3x+y–2z=33 (4) 3x–y+2z=11
Matrix Matching Questions
15. The point at which the axes are translated to remove first degree terms is
Choose the correct answer from the options given below:
(A) (B) (C) (D)
(1) s p q r
(2) p q r s
(3) s r q p
(4) p r s q
16. Match the following A (2, – 3, 4), B(–5, 1, –7). The ratio in which AB divides
List – I List – II
(A) xy-plane (p) 3 : 1
(B) yz-plane (q) 4 : 7
(C) zx-plane (r) 2 : 5
(A) (B) (C)
(1) q r p
(2) r p q
(3) q p r
(4) p q r
17. If A(cosα, sinα, 0), B(cosb, sinb, 0), C(cosg, sing, 0) are vertices of a ∆ABC and let cosα + cos b + cos g = 3a and sinα + sin b + sin g = 3b, then match the following
List – I List – II
(A) Circumcentre (p) (3a, 3b, 0)
(B) Centroid (q) (0, 0, 0)
(C) Orthocenter (r) (a, b, 0)
(s) (2a, 2b, 0)
Choose the correct answer from the options given below:
(A) (B) (C)
(1) q r s
(2) q r p
(3) p q r
(4) s r q
18. Match the items of List-I with the items of List-II choose the correct option
List – I
(A) The centroid of the triangle formed by (2, 3, –1), (5, 6, 3), (2, –3, 1) is
(B) The circumcentre of the triangle formed by (1, 2, 3), (2, 3, 1), (3, 1, 2) is
BRAIN TEASERS
List – II
(p) (2, 2, 2)
(q) (3, 1, 4)
1. Let ABCD be a tetrahedron in which the coordinates of each of its vertices are in arithmetic progression with same common difference. If the centroid G of the tetrahedron is (2, 3, k), then the distance of G from the origin is (1) 38 (2) 7
(3) 22 (4) 29
2. In a three dimensional co – ordinate system P,Q,R are images of a point A(2, 3, 5) in xy , yz and zx planes respectively. If G is the centroid of triangle PQR then area of triangle AOG is _____(O is origin) (1) 38 (2) 0 (3) 76 (4) 30
CHAPTER TEST – JEE MAIN
Section - A
1. If the points (5, 4, 2), (8, k, –7) and (6, 2, –1) are collinear, then k = (1) –2 (2) 2 (3) 10 (4) 1
2. If A = (1, 2, 3), B = (2, 3, 4) and AB is produced upto C such that 2AB = BC, then C = (1) (5, 4, 6)
(C) The orthocentre of the triangle formed by (2, 1, 5), (3, 2, 3), (4, 0, 4) is (r) (1, 1, 0)
(D) The incentre of the triangle formed by (0, 0, 0), (3, 0, 0), (0, 4, 0) is (s) (3, 2, 1) (t) (0, 0, 0)
(A) (B) (C) (D)
(1) s p q r
(2) p q r s
(3) s t q r
(4) s p t r
3. Let P r(x r,yr,zr), r = 1, 2, 3 be three points where x1, x2, x3 ; y1, y2, y3 ; z1, z2, z3 are each in G.P. with the same common ratio. Then P1, P2, P 3 are (1) coplanar (2) collinear (3) form an equilateral triangle (4) lies on a circle
4. If (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) are the vertices of an equilateral triangle such that (x1 – 2)2 + (y1 – 3)2 + (z1 – 4)2 = (x2 – 2)2 + (y2 – 3)2 + (z2 – 4)2 = (x3 – 2)2 + (y3 – 3)2 + (z3 – 4)2 then ∑x1 + 2∑y1 + 3∑z1 = ____.
(2) (6, 2, 4) (3) (4, 5, 6) (4) (6, 4, 5)
3. If the point [x1 + t(x2 – x1), y1 + t(y2 – y1 ), z1 + t(z2 – z1)] divides the line segment joining (x1, y1, z1) and (x2, y2, z2) externally, then t can be
(1) t > 0 (2) 0 < t < 1 (3) t > 1 (4) t = 1
4. A = (2, 4, 5) and B = (3, 5, –4) are two points. If the xy-plane, xz-plane divides AB in the ratio a : b, p : q respectively. Then ap bq −=
(1) 23 12 (2) 9 20
(3) 7 12 (4) 41 20
5. If h,k are the perpendicular distances from (4, 2, 5) to the x-axis, y-axis respectively, then hk =
(1) 29 (2) 13
(3) 2941 (4) 1319
6. D (2, 1, 0), E (2, 0, 0), F (0, 1, 0) are midpoints of the sides BC,CA,AB of ∆ABC respectively. Then, the centroid of ∆ABC is
(1) 111 ,, 333 (2) 42 ,,0 33
(3) 111 ,, 333
(4) 211 ,, 333
7. If (2, 1, 3), (3, 2, 5), (1, 2, 4) are the mid points of the sides BC , CA , AB of ∆ ABC respectively, then the vertex A is (1) (2, 3, 6) (2) (0, 2, 2) (3) (4, 1, 4) (4) (2, 5, 4)
8. If the centroid of the tetrahedron ABCD is 5 0,1, 2 G and three vertices are A(–2, 3, 4), B(3, –4, 2), C(2, –5, 1) then the distance of centroid G from fourth vertex D is (1) 73 4 (2) 37 2 (3) 73 2 (4) 37 4
9. If (–3, 0, 3) is a centroid of a triangle for which (3, –9, 11), (–2, 5, 7) are two vertices, then third vertex (1) (1, 2, 9) (2) (–10, 4, –9) (3) (–1, –5, –2) (4) (1, 3, –1)
10. If (2, 1, 3), (3, 1, 5) and (1, 2, 4) are the mid points of the sides BC,CA,AB or respectively, then the perimeter of the triangle is (1) 263 + (2) () 2563 ++ (3) ()263 + (4) 63 +
11. The distance between the origin and the centroid of the tetrahedron whose vertices are (0, 0, 0), (a, 0, 0), (0, b, 0), (0, 0, c) is (1) 222 abc ++ (2)
222 2 abc ++ (3) 222 4 abc ++ (4) 222 4 abc ++
12. The area of the quadrilateral PQRS with vertices P(2, 1, 1), Q(1, 2, 5), R(–2, –3, 5), S(1, –6, –7) is equal to (1) 838 (2) 938 (3) 54 (4) 738
13. α, β, γ are the roots of x3 – 2x2 – x + 2 = 0. Centroid of triangle with vertices (α, β, γ), (β, γ, α), (γ, α, β) is (1) 222 ,, 333
(3) 222 ,, 333
(2) 212 ,, 333
(4)
14. Let A(1, 2, 3), B(–1, 4, 6), C(0, –6, 4), D(1, 1, 1) be the vertices of a tetrahedron and G be the centroid of tetrahedron, G1 be the centroid of triangle BCD then 1AG AG = (1) 5 3 (2) 4 3 (3) 5 14 (4) 5 4
15. A(0, 2, 3), B(2, –1, 5), C(3, 0, –3) are vertices of ∆ABC. If a,b,c are HG,GS,SH then their increasing order is (H,G,S are orthocentre, centroid and circumcentre) (1) a,b,c (2) c,b,a (3) b,a,c (4) b,c,a
16. If (cosα, sinα, 0), (cosβ, sinβ, 0), (cosγ, sinγ, 0) are vertices of a triangle then circum radius R is
(1) 1 (2) 2 (3) 3 (4) 4
17. In the tetrahedron ABCD , A =(1, 2, –3) and G (–3, 4, 5) is the centroid of the tetrahedron. If P is the centroid of the ∆ BCD then AP =
(1) 821 3 (2) 421 3
(3) 421 (4) 21 3
18. Let A(4, 7, 8), B(2, 3, 4) and C(2, 5, 7) be the vertices of ∆ ABC . The length of the median AD is
(1) 2 (2) 1 2 (3) 77 2 (4) 89 2
19. If a ≠ 0, if the sum of the distancesof a point (a, 0, 0) and (–a, 0, 0) is a constant 2k, then the locus of that point is (1) x2 + k2(y2 + z2) = k2
(2) 22 2 222 1 xyz kka + +=
(3)
1 xyz kak + += (4) 222 2 1 1 xyzk ++= +
20. If P = (1, 2, –3), Q = (1, 0, –1) and translate the axes by Q is origin, then new coordinates of P is:
21. If A(0, 4, 1), B(a,b,c), C(4, 5, 0) and D(2, 6, 2) are the consecutive vertices of a square, then (BD)2 = _____.
22. A(2, 4, 5), B(3, 5, –4). If xy plane, yz plane divides AB in the ratio a : b and p : q, then ap bq ααβ β +=→+ = ______ ( a,b are coprime , p,q are coprime and α , b are coprime)
23. The line passing through the points (5, 1, a) and (3, b, 1) crosses yz – plane at the point 1713 0,,. 22 Then a – b = _____.
24. The x,y,z coordinates of each vertex of a triangle are in A.P. The x and y coordinates of the centroid of the triangle are 1 and 3 respectively. The distance of the centroid from the origin is k , then k = _____.
25. In ∆ ABC if 2,20,ABAC== B=(3, 2, 0) and C = (0, 1, 4). Then the length of the median passing through A is _____.
