# How do you write an equation of an ellipse in standard form given foci (0,12) and (0,-12) and a major axis of length 26?

Sep 26, 2016

${x}^{2} / 25 + {y}^{2} / 169 = 1$

#### Explanation:

Given:

${f}_{1} : \left(0 , 12\right)$

${f}_{2} : \left(0 , - 12\right)$

$M = 2 a = 26$

Note that the x-coordinate of the foci are the same. Hence we can conclude that the ellipse has a vertical major axis. The standard equation of an ellipse with a vertical major axis is

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

Get the center by getting the midpoint between the two foci,

${C}_{x} = \frac{0 + 0}{2}$

$\implies {C}_{x} = 0$

${C}_{y} = \frac{12 + - 12}{2}$

$\implies {C}_{y} = 0$

$C : \left(h , k\right) \implies \left(0 , 0\right)$

Get $c$ by getting the distance between one of the foci and the center. Since we are dealing with an ellipse with a vertical major axis, simply get the difference between the y-coordinates. For this example, let's use the first focus

$c = | {\left({f}_{1}\right)}_{y} - {C}_{y} |$

$c = 12 - 0$

$c = 12$

Get the value of $b$ using the equation

${a}^{2} - {b}^{2} = {c}^{2}$

${13}^{2} - {b}^{2} = {12}^{2}$

$\implies 169 - 144 = {b}^{2}$

$\implies {b}^{2} = 25$

$\implies b = 5$

Hence, the equation of the ellipse is

${\left(x - 0\right)}^{2} / {5}^{2} + {\left(y - 0\right)}^{2} / {13}^{2} = 1$

$\implies {x}^{2} / 25 + {y}^{2} / 169 = 1$