Find the equation of the circle passing through the points (1,2), (3,-4) and (5,-6)

Let the equation of the circle be:
$$\begin{gathered} x^2 + y^2 + 2gx + 2fy + c=0 \\ 2g + 4f + c+5=0 \\ 6g-8f+c+25=0 \\ 10g -12f+c+61=0 \end{gathered}$$
Solving we get g = - 11, f = - 2 and c = 25

Therefore, the required equation is $x^2+y^2-22x-4y+25=0$