Check the picture below.
since we know the endpoints of the diameter of the circle, then we can simply get the midpoint of those endpoints and that'd be the center,
[tex]\bf ~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ (\stackrel{x_1}{-7}~,~\stackrel{y_1}{3})\qquad (\stackrel{x_2}{5}~,~\stackrel{y_2}{1}) \qquad \left(\cfrac{ x_2 + x_1}{2}~~~ ,~~~ \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left( \cfrac{5-7}{2}~~,~~\cfrac{1+3}{2} \right)\implies \left( \cfrac{-2}{2}~~,~~\cfrac{4}{2} \right)\implies (-1~~,~~2)[/tex]
