Question

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

Foci at$\left(0,\pm 2\right)$, length of major axis is 8

Short answer

The equation of ellipse is $\frac{x^2}{12}+\frac{y^2}{16}=1$and the graph is

Step-by-step solution

Step 1. Given Information

The given data is Foci at $\left(0,\pm 2\right)$, length of major axis is 8

Step 2. Calculation

The ellipse has a center at the origin. Since focus lies on the y-axis then the y-axis is the major axis.

The distance from center to focus of the ellipse is 2 and given the length of major axis as$2a=8\Rightarrow a=4$

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

$$\begin{gathered} b^2=a^2-c^2 \\ {b}^2=16-4 \\ {b}^2=12 \end{gathered}$$

Thus, the equation of ellipse is as follows,

$$\begin{gathered} \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \\ \frac{x^2}{12}+\frac{y^2}{16}=1 \end{gathered}$$

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