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The locus of the middle points of the chords of contact of tangents to the hyperbola $\mathrm{x^2-y^2=a^2}$ drawn from points on its auxiliary circle is $\mathrm{a^2\left(x^2+y^2\right)=k\left(x^2-y^2\right)^2}$ , where $\mathrm{k=}$
Option: 1
1

Option: 2
2

Option: 3
4

Option: 4
$\frac{1}{2}$

Answers (1)

best_answer

A point on the auxiliary circle $\mathrm{x^2+y^2=a}$ is $\mathrm{P(a \cos \theta, a \sin \theta).}$

The equation of the chord of contact of tangents drawn from P to the given hyperbola is

$\mathrm{ x \cos \theta-y \sin \theta=a }$ $........(1)$

If $\mathrm{M(h, k)}$ is its middle point, then this equation can be written as $\mathrm{ T=S_1}$

Or $\mathrm{h x-k y-a^2=h^2-k^2-a^2}$ $........(2)$

On comparing (1) and (2), we obtain

$\mathrm{ \begin{aligned} \frac{\cos \theta}{h} & =\frac{-\sin \theta}{-k}=\frac{a}{h^2-k^2} \\\\ \Rightarrow \quad \cos \theta & =\frac{a h}{h^2-k^2}, \sin \theta=\frac{a k}{h^2-k^2} \end{aligned} }$

On eliminating $\mathrm{\theta, }$ we obtain

$\mathrm{ \frac{a^2 h^2}{\left(h^2-k^2\right)^2}+\frac{a^2 k^2}{\left(h^2-k^2\right)^2}=1 \Rightarrow\left(h^2+k^2\right) a^2=\left(h^2-k^2\right)^2 }$
The locus of $\mathrm{M(h, k)}$ is $\mathrm{\left(x^2+y^2\right) a^2=\left(x^2-y^2\right)^2. }$

$\mathrm{ \Rightarrow \quad k=1 }$

The answer is (a)

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