Question

In the following exercises, find the equation of the ellipse shown in the graph.

Short answer

The equation of the ellipse is$\frac{x^2}{9}+\frac{y^2}{16}=1.$

Step-by-step solution

Step 1.Given information

The graph of the ellipse is

Step 2.Concept used

The standard form of the ellipse with a center at the origin is$$\begin{gathered} \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \\ a>b\Rightarrow \text{majoraxisis}x-\text{axiswithendpoints}\left(-a,0\right)\text{,(a,0)} \\ b>a\Rightarrow \text{majoraxisis}y-\text{axis.withendpoints}\left((0,-b\right)\text{,(0,b)} \end{gathered}$$

Step 3.Find the equation of the ellipse

From the given graph center at the origin.

The major axis is on $y-$axis with endpoints$\left(0,-4\right),(0,4).$So,$b=4\Rightarrow b^2=16.$

The minor axis is on $x-$axis with endpoints$\left(-3,0\right),(3,0).$So,$a=3\Rightarrow a^2=9.$

Hence, the equation of the ellipse is

$\frac{x^2}{9}+\frac{y^2}{16}=1.$