Question

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

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

Answer 1

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

Let mid point of the chord is $P(h,k)$

Thus equation of chord with $P$ as mid point to the circle $x^2+y^2=16$ is given by,$T=S_1$

$\Rightarrow hx+ky=h^2+k^2\Rightarrow y=-\frac{h}{k}x+\frac{h^2+k^2}{k}..\left(1\right)$

Given hyperbola may be written as, $\frac{x^2}{16}-\frac{y^2}{9}=1$

Since line (1) is tangent to the hyperbola, so using condition of tangency,

$c^2=a^2{m}^2-b^2$

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

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

Hence locus of $p(h,k)$ is,$(x^2+y^2{)}^2=16x^2-9y^2$