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

Answer: (A)

Plan If $S:a{x}^2+2hxy+b{y}^2+2gx+2fy+C$
then equation of chord bisected at $P(x_1,y_1)$ is $T=S_1$
or $ax{x}_1+h(x{y}_1+y{x}_1)+by{y}_1+g(x+x_1)+f(y+y_1)+C$
$=a{x}_1^2+2h{x}_1{y}_1+b{y}_1^2+2g{x}_1+2fy,+C$
Description of Situation As equation of chord of contact is $T=0$

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

and equation of chord bisected a t the point $Q(h,k)$ is
$xh+yk-9=h^2+k^2-9$
$\Rightarrow xh+ky=h^2+k^2...(ii)$
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 $\lambda =\frac{9h}{h^2+k^2}$
$\Rightarrow \frac{20h}{4h-5k}=\frac{9h}{h^2+k^2}$
or $20(h^2+k^2)=9(4h-5k)$
or $20(x^2+y^2)=36x-45y$