Question

Find an equation

for the ellipse whose graph is shown.

Short answer

Hence, the equation of the ellipse is

Step-by-step solution

Step 1. Given information.

Given graph is centered at the origin.

Step 2. Write down the concept.

The value of both h, k are zero.

The given ellipse is a horizontal ellipse as its longer axis lies on the horizontal axis.

In given graph,

The distance from the center to the endpoint of the minor axis is given as 16 units, therefore b = 16,

Putting these values (-8 , 6) in the standard equation of an ellipse, we get

$\begin{array}{l}\Rightarrow \frac{{\left(8-0\right)}^2}{{16}^2}+\frac{{\left(6-0\right)}^2}{a^2}=1\\ \Rightarrow \frac{64}{256}+\frac{36}{a^2}=1\\ \Rightarrow \frac{4}{a^2}=1-\frac{64}{256}\\ \Rightarrow \frac{4}{a^2}=\frac{1}{12}\\ \Rightarrow a^2=48\end{array}$

Using this value of $a^2$in the standard equation we get

$\begin{array}{l}\Rightarrow \frac{x^2}{2^2}+\frac{y^2}{16/3}=1\\ \Rightarrow \frac{x^2}{4}+\frac{3y^2}{16}=1\end{array}$