Question

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

Center at $(1,2)$: focus at $(4,2)$: contains the point$(1,3)$

Short answer

The equation of the ellipse is $\frac{{(x-1)}^2}{10}+{(y-2)}^2=1$and graph of the ellipse is

Step-by-step solution

. Given information.

Center at $(1,2)$: focus at $(4,2)$: contains the point $(1,3)$

By using distance formula the value of $a$is

$$\begin{gathered} a=\sqrt{{(4-1)}^2+{(2-3)}^2} \\ a=\sqrt{9+1} \\ a=\sqrt{10} \end{gathered}$$

Since $a=\sqrt{10}$and the major axis is parallel to the x-axis, the vertices are $\sqrt{10}$units to the left and right of the center.

Thus, the vertices are $v_1=(1+\sqrt{10},2)$and $v_2=(1-\sqrt{10},2)$.

The distance between the center and the focus will give the value of $c$.

$$\begin{gathered} c=\sqrt{{(4-1)}^2+{(2-2)}^2} \\ c=\sqrt{9} \\ c=3 \end{gathered}$$

Step 2. The equation of ellipse.  

Substituting $c=3$and $a=\sqrt{10}$we get.

localid="1646723555627" $$\begin{gathered} b^2={\left(\sqrt{10}\right)}^2-{\left(3\right)}^2 \\ {b}^2=10-9 \\ {b}^2=1 \end{gathered}$$

The equation of the ellipse is$\frac{{(x-1)}^2}{10}+{(y-2)}^2=1$

.Graph of the ellipse .