Question

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

Vertices at$\left(\pm 4,0\right)$, y-intercepts are$\pm 1$

Short answer

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

Step-by-step solution

Step 1. Given Information

We are given vertices and y-intercept as$\left(\pm 4,0\right)and\pm 1$

Step 2. Calculation

The major axis of the ellipse is the x-axis and the center is at the midpoint of vertices which is the origin. From the given y-intercepts, we have, $b=1$and the distance of the center from one of the vertices is $a=4$

Find the value of foci using the formula.

$$\begin{gathered} c^2=a^2-b^2 \\ {c}^2=16-1 \\ {c}^2=15 \end{gathered}$$

Thus, the equation of ellipse is as follows,

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

On plotting the points on the graph, we get,