Question

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

Short answer

The equation of the ellipse is$\frac{x^2}{16}+\frac{y^2}{36}=1.$

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$\left(0,-6\right),(0,6).$So,$b=6\Rightarrow b^2=36.$

The minor axis is on $x-$axis with endpoints

$\left(-4,0\right),(4,0).$So,role="math" localid="1645530673898" $a=4\Rightarrow a^2=16.$

Hence, the equation of the ellipse is

$\frac{x^2}{16}+\frac{y^2}{36}=1.$