By Ib Elite Tutor at 0:04 am, May 12, 2017
Tangent and Normal 1.
Find the equations of the tangent and the normal to the following curves at the indicated points: (i)
x = θ + sin θ, y = 1 + cos θ
at θ = / 2
(ii)
2at 2 2at 3 x , y 1 t2 1 t2
at t = 1 / 2
(iii) x = at², y = 2at
at t = 1
(iv) x = a sec t, y = b tan t
at t
(v)
at θ
x = a (θ + sin θ), y = a(1 – cos θ)
Ans. : Tangent
2.
(i)
2x + 2y – – 4 = 0
2x – 2y =
(ii)
13x – 16y – 2a = 0
16 + 13y – 9a = 0
(iii) x – y + a = 0
x + y = 3a
(iv) b x sec t – a y tan t = ab
ax cos t + by cot t = (a² + b²)
(v)
(y – 2a) tan (θ/2) + x – a θ = 0
y = (x – a θ) tan (θ/2)
Find the equation of the normal to the curve x² + 2y² – 4x – 6y + 8 = 0 at the point whose abscissa is 2.
3.
Normal
[Ans. : x = 2]
Find the equation of the normal to the curve ay² = x³ at the point (am², am³). [Ans. : 2x + 3m y – am²(2 + 3m²) = 0]
4.
The equation of the tangent at (2, 3) on the curve y² = ax³ + b is y = 4x – 5. Find the values of a and b.
5.
Find the equation of the tangent line to the curve y = x² + 4x – 16 which is parallel to the line 3x – y = 1 = 0.
6.
[Ans. : a = 2, b = – 7]
[Ans. : 12x – 4y – 65 = 0]
Find an equation of normal line to the curve y = x³ + 2x + 6 which is parallel to the line x + 14y + 4 = 0.
[Ans. : x + 14y + 86 = 0, x + 14y – 254 = 0]