Question

How do you write an equation of an ellipse in standard form given Vertices: (-9,18), (-9,2), Foci (-9,14), (-9,2)?

Answer 1

The foci are `(h,k-sqrt(a^2-b^2)) and (h,k+sqrt(a^2-b^2))` for an ellipse with a vertical major axis. The vertices are `(h,k-a) and (h,k+a)`
The equation is `(y-k)^2/a^2+(x-h)^2/b^2=1" [1]"`

Explanation:

Use the given vertices to write 3 equations:

`h = -9" [2]"`
`k-a=2" [3]"`
`k+a=18" [4]"`

To find the value of k, add equations [3] and [4]:

`2k = 20`

`k = 10`

To find the value of a, substitute 10 for k into equation [4]:

`10+a=18`

`a = 8`

Use `a = 8, k = 10` and the y coordinate of one of the foci to write an equation where b is the only unknown:

`14 = 10 + sqrt(8^2 - b^2)`

`4 = sqrt(8^2 - b^2)`

`16 = 8^2 - b^2`

`-48 = -b^2`

`b = 4sqrt(3)`

Substitute, `h = -9,k=10, a = 8, and b = 4sqrt3` into equation [1]:

`(y-10)^2/8^2+(x- -9)^2/(4sqrt3)^2=1" "larr` the answer.