# How do you write an equation of an ellipse given the endpoints of major axis at (0,5) and (-0,-5) and foci at (12,0) and (-12,0)?

Jan 7, 2017

If the foci are at $\left(- 12 , 0\right) \mathmr{and} \left(12 , 0\right)$, then the endpoints $\left(0 , - 5\right) \mathmr{and} \left(0 , 5\right)$ must be on the minor axis; not the major axis. The general equation for the an ellipse with horizontally oriented foci is:

${\left(x - h\right)}^{2} / {a}^{2} + {\left(y - k\right)}^{2} / {b}^{2} = 1 \text{ [0]}$

The minor axis endpoints are $\left(h , k - b\right) \mathmr{and} \left(h , k + b\right)$
The foci are are $\left(h - \sqrt{{a}^{2} - {b}^{2}} , k\right) \mathmr{and} \left(h + \sqrt{{a}^{2} - {b}^{2}} , k\right)$

Using the points, $\left(0 , - 5\right) \mathmr{and} \left(0 , 5\right)$, we can write the following equations:

$h = 0 \text{ [1]}$
$k - b = - 5 \text{ [2]}$
$k + b = 5 \text{ [3]}$

Using the points, $\left(- 12 , 0\right) \mathmr{and} \left(12 , 0\right)$, we can write the following equations:

$k = 0 \text{ [4]}$
$h - \sqrt{{a}^{2} - {b}^{2}} = - 12 \text{ [5]}$
$h + \sqrt{{a}^{2} - {b}^{2}} = 12 \text{ [6]}$

Substitute 0 for k into equation [3]:

$b = 5$

Substitute 0 for h and 5 for b into equation [6]:

$0 + \sqrt{{a}^{2} - {5}^{2}} = 12$

${a}^{2} = 144 - 25$

$a = \sqrt{119}$

Substitute the know values into equation [0]:

${\left(x - 0\right)}^{2} / {\left(\sqrt{119}\right)}^{2} + {\left(y - 0\right)}^{2} / {5}^{2} = 1 \text{ [7]}$

Equation [7] is the one that I think that you want.