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. $36(x^2+y^2)+20x-5y=0$

Answer 1

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

Here, equation of chord of contact w.r.t P is
$x\lambda +y\left(\frac{4\lambda -20}{5}\right)=9$
$5\lambda x+(4\lambda -20)y=45\dots \dots \left(i\right)$


And equation of chord bisected at the point Q (h, k) is
$xh+yk-9=h^2+k^2-9\Rightarrow xh+ky=h^2+k^2\dots \dots \left(ii\right)$
From Eqs. (i) and (ii), we get
$\frac{5\lambda }{h}=\frac{4\lambda -20}{k}=\frac{45}{h^2+k^2}\therefore \lambda =\frac{20h}{4h-5k}and\lambda =\frac{9h}{h^2+k^2}\Rightarrow \frac{20h}{4h-5k}=\frac{9h}{h^2+k^2}or20(h^2+k^2)=9(4h-5k)or20(x^2+y^2)=36x-45y$