Question

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

Center at $(1,2)$: focus at$(1,4)$: contains the point$(2,2)$

Short answer

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

Step-by-step solution

. Given information. 

Center at $(1,2)$: focus at $(1,4)$contains the point$(2,2)$.

Since the center and the focus lie on the line $x=1$, thus the major axis is parallel to the y-axis.

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

Since, the point $(2,2)$lies on the ellipse

$$\begin{gathered} \frac{{(2-1)}^2}{b^2}+\frac{{(2-2)}^2}{a^2}=1 \\ {b}^2=1 \end{gathered}$$

Step 2. The equation of ellipse. 

Substitute $c=2$and $a=1$we get.

$$\begin{gathered} b^2=a^2-c^2 \\ {a}^2=b^2+c^2 \\ {a}^2=1+4 \\ {a}^2=5 \end{gathered}$$

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

Step 3. Graph of the ellipse .