Answer
$\Big(x-\dfrac{1}{2}\Big)^{2}+\Big(y-\dfrac{11}{2}\Big)^{2}=\dfrac{17}{2}$
Work Step by Step
Contains the points $P(2,3)$ and $Q(-1,8)$. The midpoint of the segment $PQ$ is the center of the circle.
The equation of a circle is $(x-h)^{2}+(y-k)^{2}=r^{2}$, where $(h,k)$ is the center of the circle and $r$ is its radius.
Since the fact that the midpoint of the segment $PQ$ is the center of the circle is known, find said midpoint to obtain the center of the circle:
$\Big(\dfrac{2-1}{2},\dfrac{3+8}{2}\Big)=\Big(\dfrac{1}{2},\dfrac{11}{2}\Big)$
Use the distance between two points formula to obtain the distance between the center of the circle and the point $P$. This distance represents the radius of the circle. The formula is $d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$.
$d=\sqrt{\Big(2-\dfrac{1}{2}\Big)^{2}+\Big(3-\dfrac{11}{2}\Big)^{2}}=\sqrt{\Big(\dfrac{3}{2}\Big)^{2}+\Big(-\dfrac{5}{2}\Big)^{2}}=...$
$...=\sqrt{\dfrac{9}{4}+\dfrac{25}{4}}=\sqrt{\dfrac{34}{4}}=\sqrt{\dfrac{17}{2}}$
Since now both the center and the radius are known, substitute them into the equation of a circle formula:
$\Big(x-\dfrac{1}{2}\Big)^{2}+\Big(y-\dfrac{11}{2}\Big)^{2}=\Big(\sqrt{\dfrac{17}{2}}\Big)^{2}$
$\Big(x-\dfrac{1}{2}\Big)^{2}+\Big(y-\dfrac{11}{2}\Big)^{2}=\dfrac{17}{2}$
