Question

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

Vertices at$\left(\pm 5,0\right);c=2$

Short answer

The equation of ellipse is $\frac{x^2}{25}+\frac{y^2}{24}=1$and the graph is

Step-by-step solution

Step 1. Given Information

We are given vertices as$\left(\pm 5,0\right);c=2$

Step 2. Calculation

The vertices lie on the x-axis. So, the major axis of the ellipse is x-axis and the center lies at the midpoint of vertices that is at origin.

The foci are $\left(\pm 1,0\right)$and the distance from center from one of the vertices is $a=4$. Thus, we have,

$$\begin{gathered} b^2=a^2-c^2 \\ {b}^2=25-1 \\ {b}^2=24 \end{gathered}$$

Thus, the equation of ellipse is as follows,

$$\begin{gathered} \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \\ \frac{x^2}{25}+\frac{y^2}{24}=1 \end{gathered}$$

On plotting the points on the graph, we get,