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

1 Answer
Nov 16, 2016

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