Question

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line $4x-5y=20$ to the circle $x^2+y^2=9$ is

• A. $20\left(x^2+y^2\right)-36x+15y=0$
• B. $20\left(x^2+y^2\right)+36x-15y=0$
• C. $36\left(x^2+y^2\right)-20x+45y=0$
• D. $36\left(x^2+y^2\right)+20x-45y=0$

Answer 1

Answer: (A)

Equation of the chord bisected at $P(h,k)$
$hx+ky=h^2+k^2$ (i)
Let any point on line be $\left(\alpha ,\frac{1}{5}\alpha -4\right)$
Equation of the chord of contact is
$\Rightarrow \alpha x+\left(\frac{4}{5}\alpha -4\right)y=9$
Comparing (i) and (ii)
$\frac{h}{\alpha }=\frac{k}{\frac{4}{5}\alpha -4}=\frac{h^2+k^2}{9}$
$\alpha =\frac{20h}{4h-5k}$
Now, $\frac{h(4h-5k)}{20h}=\frac{h^2+k^2}{9}$
$20\left(h^2+k^2\right)=9(4h-5k)$
$20\left(x^2+y^2\right)-36x+45y=0$