The equation of a circle:
\((x-h)^2+(y-k)^2=r^2\)
(h,k) - the coordinates of the center
r - the radius
The center is the midpoint of the diameter.
\((-3,1) \\
x_1=-3 \\ y_1=1 \\ \\
(5,7) \\
x_2=5 \\ y_2=7 \\ \\
\hbox{the midpoint: } (\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}) \\
\frac{x_1+x_2}{2}=\frac{-3+5}{2}=\frac{2}{2}=1 \\
\frac{y_1_y_2}{2}=\frac{1+7}{2}=\frac{8}{2}=4 \\
\Downarrow \\
(1,4) \\ h=1 \\ k=4\)
The radius is half the diameter, or the distance from one of the endpoints to the center.
\(r=\sqrt{(x_1-h)^2+(y_1-k)^2}=\sqrt{(-3-1)^2+(1-4)^2}=\\
=\sqrt{(-4)^2+(-3)^2}=\sqrt{16+9}=\sqrt{25}=5 \\ \Downarrow \\
r^2=5^2=25\)
The equation of the circle:
\(\boxed{(x-1)^2+(y-4)^2=25}\)
