Question

Tangents are drawn from any point on the hyperbola $4x^2-9y^2=36$ 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

Answer: 9

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

$\frac{x_1}{h}=\frac{y_1}{k}=\frac{9}{h^2+k^2}$
$x_1=\frac{9h}{h^2+k^2}=3\sec \theta$
$y_1=\frac{9k}{h^2+k^2}=2\tan \theta$
On using ${\sec }^2\theta -{\tan }^2\theta =1$, we get
${\left(\frac{3h}{h^2+k^2}\right)}^2-{\left(\frac{9k}{2\left(h^2+k^2\right)}\right)}^2=1$
$\Rightarrow \frac{81x^2}{9}-\frac{81y^2}{4}={\left(x^2+y^2\right)}^2\Rightarrow \frac{x^2}{9}-\frac{y^2}{4}={\left(\frac{x^2+y^2}{9}\right)}^2$
$\therefore k=9$