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. $x^2+y^2=7$
• B. ${\left(x^2+y^2\right)}^2=7$
• C. ${\left(x^2+y^2\right)}^2=16x^2-9y^2$
• D. ${\left(x^2+y^2\right)}^2=25$

Answer 1

Answer: (C)

Let $(h,k)$ be the mid point of the chord of the circle $x^2+y^2=16$, so that its equation by $T=S_1$ is $hx+ky=h^2+k^2$
or $y=-\frac{h}{k}x+\frac{h^2+k^2}{k}$
i.e. the form $y=mx+c$
It will touch the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$
if $c^2=16m^2-9$
$\therefore {\left(\frac{h^2+k^2}{k}\right)}^2=16{\left(-\frac{h}{k}\right)}^2-9$
${\left(h^2+k^2\right)}^2=16h^2-9k^2$
Generalising, the locus of the mid-point $(h,k)$ is ${\left(x^2+y^2\right)}^2=16x^2-9y^2$