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. $20(x^2+y^2)-36x+45y=0$
• B. $20(x^2+y^2)+36x-45y=0$
• C. $36(x^2+y^2)-20y+45y=0$
• D. $36(x^2+y^2)+20x-45y=0$

Answer 1

The correct option is A $20(x^2+y^2)-36x+45y=0$


Any point on the line $4x-5y=20$ is $P\equiv (\alpha ,\frac{4\alpha -20}{5}).$

Let mid point of the chord of contact drawn from point $P$ to the circle $x^2+y^2=9$ is $Q(h,k).$

Equation of chord $AB$ is $T=0$
$x\alpha +y\left(\frac{4\alpha -20}{5}\right)=9$
$\Rightarrow 5\alpha x+(4\alpha -20)y=45\dots \dots \left(i\right)$

And equation of chord $AB$ bisected at the point $Q(h,k)$ is $T=S_1$
$xh+yk-9=h^2+k^2-9\Rightarrow xh+ky=h^2+k^2\dots \dots \left(ii\right)$

As line $\left(i\right)$ and $\left(ii\right)$ represent the same line.
$\therefore \frac{5\alpha }{h}=\frac{4\alpha -20}{k}=\frac{45}{h^2+k^2}$
$\Rightarrow \alpha =\frac{20h}{4h-5k}$
and $\alpha =\frac{9h}{h^2+k^2}$
$\Rightarrow \frac{20h}{4h-5k}=\frac{9h}{h^2+k^2}\Rightarrow 20(h^2+k^2)=9(4h-5k)$

Hence, required locus is
$20(x^2+y^2)-36x+45y=0$