Circle

Equation of circle passes through (3,5),(5,5) and (3,−3) is
Let equation be
x2+y2+2gx+2fy+c=0
This is passes through (3,5),(5,5) and (3,−3)
6g+10f+c=−34
10g+10f+c=50
6g−6f+c=−18
From Eqs. (i), (ii) and (iii) g=−4,
f=−1‌ and ‌c=0
Required equation of circle
x2+y2−8x−2y=0
and given circle is
x2+y2+2x+2fy=0
are orthogonal
2(−4)(1)+2(−1)f‌=0+0
f‌=−4