Question

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

Foci at$(0,\pm 3)$, x-intercepts are$\pm 2$

Short answer

The equation of ellipse is $\frac{x^2}{4}+\frac{y^2}{13}=1$and the graph is

Step-by-step solution

Step 1. Given Information

We are given foci and x-intercept as$(0,\pm 3);\pm 2$respectively.

Step 2. Calculation

The center is at the midpoint of foci which is the origin. The major axis of the ellipse is the y-axis and the distance between the given x-intercept is less than the distance between the foci.

The x-intercepts of the ellipse are $(b,0)and(-b,0)$. So, $b=\pm 2$and since one of the foci is $(0,3)$. So, $c=3$

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

$$\begin{gathered} b^2=a^2-c^2 \\ 4=a^2-9 \\ {a}^2=13 \end{gathered}$$

Thus, the equation of ellipse is as follows,

$$\begin{gathered} \frac{x^2}{b^2}+\frac{y^2}{a^2}=1 \\ \frac{x^2}{4}+\frac{y^2}{13}=1 \end{gathered}$$

On plotting the points on the graph, we get,