Question

How do you find an equation for the ellipse with vertices at (-6,4) and (10,4); focus at (8,4)?

Answer 1

Please see the explanation.

Explanation:

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 +-sqrt(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:

`2h = 4`
`h = 2`
`a = 8`

Use the focus to write another equation:

`8 =h + sqrt(a^2 - b^2)`

Substitute values for h and a:

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

`6 = sqrt(64 - b^2)`

`36 = 64 - b^2`

`b^2 = 64 - 36`

`b^2 = 28`

`b = sqrt(28)`

Substitute the values into the standard form:

`(x - 2)^2/8^2 + (y - 4)^2/(sqrt(28))^2 = 1`