The midpoint of chord formed by the polar of (–9,12) w.r.t ${\mathrm{x}}^2+{\mathrm{y}}^2=100$ is

Equation of  polar of  (-9,12) is S₁ = 0.
$$\begin{gathered} x\left(-9\right)+y\left(12\right)=100 \\ \Rightarrow x\left(9\right)-y\left(12\right)+100=0 \end{gathered}$$
Midpoint is foot of the $\perp$ ar from the centre on the chord
$$\begin{gathered} \frac{h}{9}=\frac{k}{-12}=\frac{-100}{81+144} \\ \Rightarrow \left(h,k\right)=\left(-4,\frac{16}{3}\right) \end{gathered}$$