Question

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

Center at $(-3,1)$: vertex at $(-3,3)$: focus at$(-3,0)$

Short answer

The equation of the ellipse is $\frac{{(x+3)}^2}{3}+\frac{{(y-1)}^2}{4}=1$graph of the ellipse is

Step-by-step solution

Step 1. Given information.

Center at $(-3,1)$: Vertex at $(-3,3)$: focus at$(-3,0)$.

Since the center, focus and vertex all lies on the line $x=-3$the major axis is parallel to the y- axis.

The distance from the center of the ellipse to the focus is $c=1$

The distance from the center of the ellipse to the vertex is$a=2$

Step 2. The equation of ellipse. 

Substitute $c=1$and $a=2$in $b^2=a^2-c^2$we get.

$$\begin{gathered} b^2=a^2-c^2 \\ {b}^2=2^2-1^2 \\ {b}^2=4-1 \\ {b}^2=3 \end{gathered}$$

The equation of the ellipse is$\frac{{(x+3)}^2}{3}+\frac{{(y-1)}^2}{4}=1$

Step 3. Graph of the ellipse.