We need to calculate a system of equations using the general formula of the circle
[tex]x^2+y^2+Dx+Ey+F=0[/tex]for the point (-5, 5)
x=-5
y=5
the first equation is
[tex]\begin{gathered} (-5)^2+(5)^2-5D+5E+F=0 \\ 25+25-5D+5E+F=0 \\ 50-5D+5E+F=0 \\ -5D+5E+F=-50 \end{gathered}[/tex]for the point (-5,-3)
x=-5
y=-3
the second equation is
[tex]\begin{gathered} (-5)^2+(-3)^2-5D-3E+F=0 \\ 25+9^{}-5D-3E+F=0 \\ 34^{}-5D-3E+F=0 \\ -5D-3E+F=-34 \end{gathered}[/tex]for the point (7, -3)
x=7
y=-3
the third equation is
[tex]\begin{gathered} (7)^2+(-3)^2+7D-3E+F=0 \\ 49+9+7D-3E+F=0 \\ 58+7D-3E+F=0 \\ 7D-3E+F=-58 \end{gathered}[/tex]Then we solve the system of equations, and we obtain
D=-2
E=-2
F=50
The equation of the circle that passes through these points is
[tex]x^2+y^2-2x-2y-50=0[/tex]for calculate the coordinates of the center of the circle we have
D=-2h
h is the x coordinate of the center of the circle
D=-2
we isolate the h
h=-2/-2=1
E=-2k
k is the y coordinate of the center of the circle
E=-2
we isolate the k
k=-2/-2=1
the center of the circle is (h,k)=(1,1)
