Question

Tangents are drawn from any point on the hyperbola $4 x^2-9 y^2=36$ to the circle $x^2+y^2-9=0$. The locus of mid-point of chord of contact is $\left(\frac{x^2}{9}-\frac{y^2}{4}\right)=\left(\frac{x^2+y^2}{k}\right)^2$, where $k \in N$. Find $k$.

Answer 1

$H : \frac{ x ^2}{9}-\frac{ y ^2}{4}=1$
Equation of AB:
$xx _1+ yy _1=9 $ ....(1) (Chord of contact)
$xh + yk = h ^2+ k ^2$ .....(2) $(T = S_1)$
On comparing equation (1) and (2), we get

$\frac{ x _1}{ h }=\frac{ y _1}{ k }=\frac{9}{ h ^2+ k ^2} $
$x _1=\frac{9 h }{ h ^2+ k ^2}=3 \sec \theta $
$y _1=\frac{9 k }{ h ^2+ k ^2}=2 \tan \theta$
On using $\sec ^2 \theta-\tan ^2 \theta=1$, we get
$\left(\frac{3 h }{ h ^2+ k ^2}\right)^2-\left(\frac{9 k }{2\left( h ^2+ k ^2\right)}\right)^2=1$
$\Rightarrow \frac{81 x^2}{9}-\frac{81 y^2}{4}=\left(x^2+y^2\right)^2 \Rightarrow \frac{x^2}{9}-\frac{y^2}{4}=\left(\frac{x^2+y^2}{9}\right)^2$
$\therefore k =9$