Question

Find an equation for each ellipse. Graph the equation by hand.

Focus at$\left(0,-4\right)$, vertices at$\left(0,\pm 8\right)$

Short answer

The equation of ellipse is $\frac{x^2}{48}+\frac{y^2}{64}=1$and the graph is

Step-by-step solution

Step 1. Given Information

We are given Focus and vertices as$\left(0,-4\right)and(0,\pm 8)$respectively.

Step 2. Calculation

Since the given vertices and focus lies on the y-axis then the y-axis is the major axis. The center is the midpoint of the vertices

The distance from the center to one of the foci is 4 and the distance from the center to one of the vertices is 8.

Find the value of length of minor axis using the formula.

$$\begin{gathered} b^2=a^2-c^2 \\ {b}^2=64-16 \\ {b}^2=48 \end{gathered}$$

Thus, the equation of ellipse is as follows,

$$\begin{gathered} \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \\ \frac{x^2}{48}+\frac{y^2}{64}=1 \end{gathered}$$

On plotting the vertices and foci on the graph, we get,