The midpoint of the line segment connecting the endpoints of the diameter is the center of the circle. We can find the midpoint by averaging the coordinates of the endpoints:
$\sf\implies\:Midpoint\:=\:\left(\frac{-6\:-\:14}{2},\:\frac{3\:+\:13}{2}\right)\:=\:(-10,\:8)$
The radius of the circle is half the length of the diameter, which we can find using the distance formula:
$\sf\implies\:Radius\:=\:\frac{\sqrt{(-6\:-\:(-14))^2\:+\:(3\:-\:13)^2}}{2}\:=\:\frac{\sqrt{160}}{2}\:=\:4\sqrt{10}$
Thus, the equation of the circle is
$\sf\implies\:(x\:+\:10)^2\:+\:(y\:-\:8)^2\:=\:(4\sqrt{10})^2$
Simplifying and rearranging, we get:
$\sf\implies\red\bigstar\:(x\:+\:10)^2\:+\:(y\:-\:8)^2\:=\:160$
