Question

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

Vertices at$(2,5)$and$(2,-1)$:c=2

Short answer

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

Step-by-step solution

Step 1. Given information.  

Vertices at $(2,5)$and $(2,-1)$: $c=2$

The vertices lies on the line $x=2$, thus the major axis is parallel to the y-axis.

The center is at the mid point of the vertices $(2,2)$.

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

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

Step 2. The equation of ellipse.  

Substitute $a=3$and $c=2$in $b^2=a^2-c^2$we get.

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

Substituting the values of $a^2$and $b^2$in the equation of the ellipse we get.

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

.Graph of the ellipse .