The equation of the required circle is, x2+y2+2gx+2fy+c=0 The circle cuts, x2+y2−2x+6y=0,x2+y2−4x−2y+6=0 and, x2+y2−12x+2y+3=0 orthogonally. Then, −2g+6f=c −4g−2f=c+6 −12g+2f=c+3 Solve the above three equations for f,g,c g=0 f=−
3
4
c=−
9
2
The equation of the required circle is x2+y2−
3
2
y−
9
2
=0 Therefore, the equation of tangent at point (0,3) is, 3y−