10.3
Ellipses
JMerrill, 2010

General Second Degree Equation

Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0

Definition - Ellipse

An ellipse is the set of all points in a plane, the sum of whose distances from two distinct fixed points (foci) is constant.

Standard Equations of Ellipses

Writing Equations from
Descriptions


Write an equation of the ellipse whose vertices are at (-3, 0) and (3, 0) and whose minor axis length is 4. Find the foci.
You Try: x
2 y
2
 
9 4
1
Foci ( 
5 0

Write an equation of the ellipse whose vertices are (-5, 0), (5, 0) and whose covertices are (0, -3), (0, 3).Find the foci.
x
2 y
2
 
25 9
1
Foci 

Writing Equations from Graphs
You Try:
 x 
3
 2
16

 y 
3

2
36

1
 x 
1
 2
81

 y 
4

2
16

1

Writing the Equation in Standard
Form & Graph

Sketch the ellipse given by x 2 + 4y 2 + 6x – 8y + 9 = 0

In order to put this in standard form, you must complete the square:
1.
Move the 9 to the right
2.
Group the x’s. Group the y’s.
3.
You MUST have a leading coefficient of 1. If it’s not 1, factor out the coefficient.
4.
Complete the square
5.
The equation must = 1; divide

You Try

Put the ellipse 4x 2 + y 2 - 8x + 4y – 8 = 0 into standard form
 x 
1

2
4

 y 
2

2
16

1

Application – You Try

A skating park has a track shaped like an ellipse. If the length of the track is 58m and the width of the track is 38m, find the equation of the ellipse.
x
2 y
2
 
841 361
1

Eccentricity
To measure the ovalness of an ellipse, we use eccentricity: