jmc

Use the angle bisector theorem to find that the angle bisector of $\angle P$ divides $QR$ into segments of length $\frac{25}{x}=\frac{15}{20-x}⟹x=\frac{25}{2},\frac{15}{2}$. It follows that $\frac{Q{P}^{\prime }}{R{P}^{\prime }}=\frac{5}{3}$, and so $P^{\prime }=\left(\frac{5x_R+3x_Q}{8},\frac{5y_R+3y_Q}{8}\right)=(-5,-23/2)$. The desired answer is the equation of the line $P{P}^{\prime }$. $P{P}^{\prime }$ has slope $\frac{-11}{2}$, from which we find the equation to be $11x+2y+78=0$. Therefore, $a+c=\boxed{89}$.