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Tangents are drawn from any point on hyperbola $4x^2 -9y^2 = 36$ to the circle $x^2 + y^2 = 9$ . If locus of midpoint of chord of contact is $\left( {\frac{{{x^2}}}{9} - \frac{{{y^2}}}{4}} \right) = \lambda {\left( {\frac{{{x^2} + {y^2}}}{9}} \right)^2}$ , then $\lambda $ is
$4$
$1$
$2$
$3$
Solution
$\frac{x^{2}}{9}-\frac{y^{2}}{4}=1 ; P(3 \sec \theta, 2 \tan \theta)$
$\therefore$ equation of chord of contact $AB$ wrt $P$ is $T$
$=0$
i.e $3 x \sec \theta+2 y \tan \theta=9$ …..$(2)$
Also, chord with given midpoint $(\mathrm{h}, \mathrm{k})$ is $T=S_{1}$
$\Rightarrow h x+k y-9=h^{2}+k^{2}-9$ …..$(2)$
comparing $( 1 )$ and $( 2 )$
$\frac{3 \sec \theta}{h}=\frac{2 \tan \theta}{h}=\frac{9}{h^{2}+k^{2}} ;$ as $\sec ^{2} \theta-\tan ^{2} \theta=1$
$\Rightarrow$ locus is $\left(\frac{x^{2}}{9}-\frac{y^{2}}{4}\right)=1\left(\frac{x^{2}+y^{2}}{9}\right)^{2}$
