Question: A line $L_1$ defined by $y =m_1x+c_1$ intersects the circle $S$ defined by $x^2+y^2+2Fx+2Gy+C$ at $P$ and $Q$. The line is then rotated about a point $(a,b)$ lying on $L_1$ to form $L_2$, defined by $y=m_2x+c_2$, which intersects $S(x,y)$ at $R$ and $S$. If the combined equations of lines $PR$ and $QS$ is $ax^2+by^2+2hxy+2gx+2fy+c$, find the equation of the circle.
Note: Everything except $F$, $G$, and $C$ is given. I only want to learn how to approach this problem. It will be especially helpful if you could generalize it.