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. ab 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=