Question

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

Foci at $\left(\pm 2,0\right)$, length of major axis is 6

Short answer

The equation of ellipse is $\frac{x^2}{9}+\frac{y^2}{5}=1$and graph is

Step-by-step solution

Step 1. Given Information

The given data is Foci at $\left(\pm 2,0\right)$, length of major axis is 6

Step 2. Calculation

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

The distance from center to focus of the ellipse is $c=2$and given the length of major axis as $2a=6\Rightarrow a=3$

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

$$\begin{gathered} b^2=a^2-c^2 \\ {b}^2=3^2-2^2 \\ {b}^2=9-4 \\ {b}^2=5 \\ b=\pm \sqrt{5} \end{gathered}$$

Thus, the equation of ellipse is as follows,

$$\begin{gathered} \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \\ \frac{x^2}{9}+\frac{y^2}{5}=1 \end{gathered}$$

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