College Algebra, Chapter 8, 8.2, Section 8.2, Problem 40

Determine the equation for the ellipse with end points of minor axis $(0, \pm 3)$ and distance between foci $8$.
The equation $\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is an ellipse that has endpoints on major axis at
$(0,\pm a)$ and endpoints on minor axis at $(0,\pm b)$ with foci on $(0,\pm c)$ where $c^2 = a^2 - b^2$ and the distance
between the foci is determined as $2c$. So, $b = 3$ and if $2c = 8$, then $c = 4$. Thus,

$\begin{equation} \begin{aligned} c^2 &= a^2 - b^2 \\ \\ a^2 &= c^2 + b^2\\ \\ a^2 &= 4^2 + 3^2 \\ \\ a^2 &= 25\\ \\ a &= 5 \end{aligned} \end{equation}$

Therefore, the equation is
$\displaystyle \frac{x^2}{3^2} + \frac{y^2}{5^2} = 1 \text{ or } \frac{x^2}{9} + \frac{y^2}{25} = 1$