Mathematics - Circle Question with Solution | TestHub

Let $P\left(t,\frac{4t-20}{5}\right)$ be a point on the line $4x-5y=20$. Then the chord of contact of tangents drawn from $P$ to the circle $x^2+y^2=9$ is $tx+\left(\frac{4t-20}{5}\right)y=9\dots (i)$ Let $\theta (h,k)$ be the mid-point of this chord of contact, then its equation is also $hx+ky=h^2+k^2\dots (ii)$ $[$ using $T=S^{\prime }]$ Clearly Eqs. (i) and (ii) represent the same line $\begin{array}{rl}& \therefore \frac{t}{h}=\frac{4t-20}{5k}=\frac{9}{h^2+k^2}\\ & \Rightarrow \frac{t}{h}=\frac{9}{h^2+k^2}\text{ and }\frac{t}{h}=\frac{4t-20}{5k}\\ & \Rightarrow t=\frac{9h}{h^2+k^2}\text{ and }t=\frac{20h}{4h-5k}\\ & \Rightarrow \frac{9h}{h^2+k^2}=\frac{20h}{4h-5k}\\ & \Rightarrow h\left\{20\left(h^2+k^2\right)-36h+45k\right\}=0\\ & x=0,\text{ or }\left[20\left(x^2+y^2\right)-36x+45y\right]=0\end{array}$