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11.2 – The Ellipse  An ellipse is the set of all points in a plane whose distances from two fixed points (called the foci) is a constant sum. (demonstration) Standard Form of an Ellipse The equations must be equal to 1. ab HORIZONTAL VERTICAL

 x  h

2

a2

y  k

 x  h

2

b2

2

 1

b2

Major Axis is parallel to the x-axis.

y  k

2

a2

 1

Major Axis is parallel to the y-axis.

Center: C  h, k 

“a” is the distance from the Center to the Major Axis Vertices. The length of the Major Axis = 2a “b” is the distance from the Center to the Minor Axis Vertices. The length of the Minor Axis = 2b “c” is the distance from the Center to the Foci. The distance between the Foci = 2c

c2  a2  b2

c  a2  b2

Vertices: Horizontal

V  h  a, k  V  h, k  b  F  h  c,k 

Vertical On the Major Axis: On the Minor Axis:

Foci:

V  h, k  a  V  h  b, k  F  h, k  c 

Remember: the foci lie inside the curve and on the Major Axis!

 In ellipses, a  b , so a2  b2 , and in the Standard Form equation, the variable (x or y) under which a 2 is located determines in which direction the ellipse is stretched!


EXAMPLES: 2 2 1. Graph 25 x  3  16  y  2  400 .

2. Write the equation of the ellipse whose semi-major axis is 6 units and the foci are at  0,2 and  8,2 .

General Form of an equation of an Ellipse: Ax2  By2  Cx  Dy  E  0 2

Note: the x and

y 2 terms are both positive, but A and B are different!

3. Write the equation of the ellipse in Standard Form and then graph the ellipse: x2  4y2  2x  16y  1  0 a=

b=

Major Axis Vertices: Minor Axis Vertices: Foci:

Center: c=


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