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

Jan 28, 2017

The foci are $\left(h , k - \sqrt{{a}^{2} - {b}^{2}}\right) \mathmr{and} \left(h , k + \sqrt{{a}^{2} - {b}^{2}}\right)$ for an ellipse with a vertical major axis. The vertices are $\left(h , k - a\right) \mathmr{and} \left(h , k + a\right)$
The equation is ${\left(y - k\right)}^{2} / {a}^{2} + {\left(x - h\right)}^{2} / {b}^{2} = 1 \text{ [1]}$

#### Explanation:

Use the given vertices to write 3 equations:

$h = - 9 \text{ [2]}$
$k - a = 2 \text{ [3]}$
$k + a = 18 \text{ [4]}$

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

$2 k = 20$

$k = 10$

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:

$14 = 10 + \sqrt{{8}^{2} - {b}^{2}}$

$4 = \sqrt{{8}^{2} - {b}^{2}}$

$16 = {8}^{2} - {b}^{2}$

$- 48 = - {b}^{2}$

$b = 4 \sqrt{3}$

Substitute, $h = - 9 , k = 10 , a = 8 , \mathmr{and} b = 4 \sqrt{3}$ into equation [1]:

${\left(y - 10\right)}^{2} / {8}^{2} + {\left(x - - 9\right)}^{2} / {\left(4 \sqrt{3}\right)}^{2} = 1 \text{ } \leftarrow$ the answer.