Question

The locus of the mid-points of the chords of the circle $x^2+y^2=16$ which are tangent to the hyperbola $9x^2-16y^2=144$, is

• A. ${(x^2+y^2)}^2=16x^2+9y^2$
• B. ${(x^2-y^2)}^2=16x^2-9y^2$
• C. ${(x^2+y^2)}^2=16x^2-9y^2$
• D. None of the above

Answer 1

The correct option is C ${(x^2+y^2)}^2=16x^2-9y^2$

Equation of chord of contact when mid point is given is $T=S^{\prime }$

Let $(h,k)$ be the mid point of chord of contact of circle $x^2+y^2=16$

So the equation of chord of contact is

$hx+ky=h^2+k^{2}ky=-hx+h^2+k^{2}y=-\frac{h}{k}x+\frac{h^2+k^2}{k}.....\left(i\right)$

Given hyperbola is

$\frac{x^2}{16}-\frac{y^2}{9}=\mathrm{1......}\left(ii\right)$

A line $y=mx+c$ is tangent to hyperbola if $c^2=a^2{m}^2-b^2........\left(iii\right)$

Comparing with $\left(i\right)$ and $\left(ii\right)$ and substituting values in $\left(iii\right)$

$\Rightarrow {\left(\frac{h^2+k^2}{k}\right)}^2=16{(-\frac{h}{k})}^2-9\Rightarrow {(h^2+k^2)}^2=16h^2-9k^2$

Generalising the equation

$\Rightarrow {(x^2+y^2)}^2=16x^2-9y^2$

So option $C$ is coorect.