Mathematics - Circle Question with Solution | TestHub

Any point $P$ on line $4x-5y=20$ can be considered as $\left(\alpha ,\frac{4\alpha -20}{5}\right)$. Equation of chord of contact $AB$ 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. $\left(\frac{4\alpha -20}{5}\right)=\mathrm{9...}(i)$
Also the equation of chord $AB$ whose mid point is $(h,k)$ is $hx+ky=h^2+k^2....(ii)$ $\because$ Equations (i) and (ii) represent the same line, $\therefore \frac{h}{\alpha }=\frac{k}{\frac{4\alpha -20}{5}}=\frac{h^2+k^2}{9}$ $\Rightarrow 5k\alpha =4h\alpha -20h$ and $9h=\alpha \left(h^2+k^2\right)$ $\Rightarrow \alpha =\frac{20h}{4h-5k}$ and $\alpha =\frac{9h}{h^2+k^2}$ $\Rightarrow \frac{20h}{4h-5k}=\frac{9h}{h^2+k^2}\Rightarrow 20\left(h^2+k^2\right)=9(4h-5k)$ $\therefore$ Locus of $(h,k)$ is $20\left(x^2+y^2\right)-36x+45y=0$