The locus of the mid-point of the chord of contact of tangents...

Any point $P$ on line $4 x-5 y=20$ can be considered as \left(\alpha, \frac{4 \alpha-20}{5}\right)$. Equation of chord of contact$A B$to the circle$x^{2}+y^{2}=9$drawn from point$\mathrm{P}\left(\alpha, \frac{4 \alpha-20}{5}\right)$is x.$\alpha+$y.$\left(\frac{4 \alpha-20}{5}\right)=9...(i)$<img src="https://cdn-question-pool.Quantrex.app/pyq/jee_advanced/zkFIWBdkdHID8Ep1HUGZDpd4kkJTYXASn79_Szm3a1E.original.fullsize.png"><br> Also the equation of chord$A B$whose mid point is$(h, k)$is$h x+k y=h^{2}+k^{2}....(ii) \because \quad$Equations (i) and (ii) represent the same line,$\therefore \quad \frac{h}{\alpha}=\frac{k}{\frac{4 \alpha-20}{5}}=\frac{h^{2}+k^{2}}{9} \Rightarrow 5 k \alpha=4 h \alpha-20 h$and$9 h=\alpha\left(h^{2}+k^{2}\right)$\Rightarrow \alpha=\frac{20 h}{4 h-5 k} \quad$ and $\alpha=\frac{9 h}{h^{2}+k^{2}}$ $\Rightarrow \quad \frac{20 h}{4 h-5 k}=\frac{9 h}{h^{2}+k^{2}} \Rightarrow 20\left(h^{2}+k^{2}\right)=9(4 h-5 k)$ $\therefore \quad$ Locus of $(h, k)$ is $20\left(x^{2}+y^{2}\right)-36 x+45 y=0$