Question

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

Foci at$(1,2)$and$(-3,2)$: vertex at$(-4,2)$.

Short answer

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

Step-by-step solution

Step 1. Given information.

Foci at $(1,2)$and $(-3,2)$: vertex at $(-4,2)$.

The focus and the vertex lie on the line $y=2$, so the major axis is parallel to the x-axis.

The center is at the midpoint of the foci$(h,k)=(-1,2)$

Step 2. The equation of ellipse.   

The distance of the center from one of the foci is $c=2$.

The distance of the center from one of the vertices is $a=3$

So, we have

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

Substitute the value of $a^2$and $b^2$in the equation of ellipse, we get

$\frac{{(x+1)}^2}{9}+\frac{{(y-2)}^2}{5}=1$

.Graph of the ellipse