Question

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

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

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 $\left(\alpha, \frac{4}{5} \alpha-4\right)$
Equation of the chord of contact is
$\Rightarrow \alpha x+\left(\frac{4}{5} \alpha-4\right) y=9 \ldots$ (ii)
Comparing (i) and (ii)
$\frac{h}{\alpha}=\frac{k}{\frac{4}{5} \alpha-4}=\frac{h^{2}+k^{2}}{9}$
$\alpha=\frac{20 h}{4 h-5 k}$
Now, $\frac{h(4 h-5 k)}{20 h}=\frac{h^{2}+k^{2}}{9}$
$20\left( h ^{2}+ k ^{2}\right)=9(4 h -5 k )$
$20\left(x^{2}+y^{2}\right)-36 x+45 y=0$