Question

The locus of the midpoints of the chords of the circle ${x^2+y}^2=16$ which are tangents to the hyperbola $9{x^2-16y}^2=144$ is

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

Answer 1

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

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