Find the equation of the ellipse with the following properties:

Vertices at (-6,-3) and (0,-3)

e=1/3

Answer: The equation of an ellipse:

(

x

-

h

)

2

a

2

+

(

y

-

k

)

2

b

2

=

1

;

a

>

b

Has vertices at

(

h

±

a

,

k

)

Has foci at

(

h

±

√

a

2

-

b

2

,

k

)

Use the vertices to write 3 equations:

k

=

4

[1]

h

-

a

=

-

6

[2]

h

+

a

=

10

[3]

Use equations [2] and [3] to solve for h and a:

2

h

=

4

h

=

2

a

=

8

Use the focus to write another equation:

8

=

h

+

√

a

2

-

b

2

Substitute values for h and a:

8

=

2

+

√

8

2

-

b

2

6

=

√

64

-

b

2

36

=

64

-

b

2

b

2

=

64

-

36

b

2

=

28

b

=

√

28

Substitute the values into the standard form:

(

x

-

2

)

2

8

2

+

(

y

-

4

)

2

(

√

28

)

2

=

1