Answer
$(x-0.5)^2+(y-5.5)^2=8.5$
Work Step by Step
We are given the points:
$$\begin{align*}
&P(2,3)\\
&Q(-1,8).
\end{align*}$$
The equation of a circle with center $(h,k)$ and radius $r$ is:
$$\begin{align}(x-h)^2+(y-k)^2=r^2.\end{align}\tag1$$
First we determine the coordinates of the center $C(h,k)$ by calculating the coordinates of the midpoint of $PQ$:
$$\begin{align*}
x_C&=h=\dfrac{x_P+x_Q}{2}=\dfrac{2+(-1)}{2}=0.5\\
y_C&=k=\dfrac{y_P+y_Q}{2}=\dfrac{3+8}{2}=5.5.
\end{align*}$$
So the center is $C(0.5,5.5)$.
We determine the radius $r$ calculating the distance between the center $C$ and one of the points $P$, $Q$:
$$\begin{align*}
r&=CP\\
&=\sqrt{(x_P-x_C)^2+(y_P-y_C)^2}\\
&=\sqrt{(2-0.5)^2+(3-5.5)^2}\\
&=\sqrt{8.5}.
\end{align*}$$
Substitute the values of $h$, $k$ and $r^2$ in Eq. $(1)$ to find the equation of the circle:
$$(x-0.5)^2+(y-5.5)^2=8.5.$$
