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  • Tangents are drawn from a point P to the hyperbola If the chord of

Tangents are drawn from a point P to the hyperbola \mathrm{x^2-y^2=a^2.} If the chord of contact of tangents is a normal, the locus of P is \mathrm{\frac{1}{x^2}-\frac{1}{y^2}=\frac{\alpha}{a^2}}, then \mathrm{\alpha=} ____________.

Option: 1

4


Option: 2

3


Option: 3

5


Option: 4

0


Answers (1)

best_answer

The chord of contact of tangents from P\left(x_1, y_1\right) is

\begin{aligned} & S_1=0 . \\ & x_1 x-y_1 y=a^2 . \end{aligned}

It is same as the normal at \theta, \frac{x}{\sec \theta}+\frac{y}{\tan \theta}=2 a

\begin{aligned} & \therefore x_1 \sec \theta=-y_1 \tan \theta=\frac{a}{2} \\ & \Rightarrow \sec \theta=\frac{a}{2 x_1}, \tan \theta=-\frac{a}{2 y_1} . \end{aligned}

Eliminating \theta, \frac{a^2}{4}\left(\frac{1}{x_1^2}-\frac{1}{y_1^2}\right)=1

\therefore  The locus of P is \frac{1}{x^2}-\frac{1}{y^2}=\frac{4}{a^2}.

Posted by

manish painkra

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