# How do you find the equation of an ellipse with vertices (0,+-8) and foci (0, +-4)?

Dec 30, 2016

${x}^{2} / 48 + {y}^{2} / 64 = 1$

#### Explanation:

Find the equation of an ellipse with vertices $\left(0 , \pm 8\right)$ and foci $\left(0 , \pm 4\right)$.

The equation of an ellipse is ${\left(x - h\right)}^{2} / {a}^{2} + {\left(y - k\right)}^{2} / {b}^{2} = 1$ for a horizontally oriented ellipse and ${\left(x - h\right)}^{2} / {b}^{2} + {\left(y - k\right)}^{2} / {a}^{2} = 1$ for a vertically oriented ellipse.

$\left(h , k\right)$ is the center and the distance $c$ from the center to the foci is given by ${a}^{2} - {b}^{2} = {c}^{2}$. $a$ is the distance from the center to the vertices and $b$ is the distance from the center to the co-vertices.

The center of the ellipse is half way between the vertices. Thus, the center $\left(h , k\right)$ of the ellipse is $\left(0 , 0\right)$ and the ellipse is vertically oriented.

$a$ is the distance between the center and the vertices, so $a = 8$.
$c$ is the distance between the center and the foci, so $c = 4$

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

${b}^{2} = {8}^{2} - {4}^{2} = 64 - 16 = 48$

The equation is:

${\left(x - 0\right)}^{2} / 48 + {\left(y - 0\right)}^{2} / 64 = 1$ or ${x}^{2} / 48 + {y}^{2} / 64 = 1$