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Q.

A tangent to the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ meets the transverse axis and conjugate axis in L and M. A rectangle CLPM is completed, where C is the centre, then locus of P is ----

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a

$\frac{a^{2}}{x^{2}} + \frac{b^{2}}{y^{2}} = 1$

b

$\frac{a^{2}}{x^{2}} - \frac{b^{2}}{y^{2}} = 1$

c

$\frac{a^{2}}{x^{2}} - \frac{b^{2}}{y^{2}} = 4$

d

$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$

answer is D.

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Detailed Solution

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Tangent is $\frac{x}{a} \sec θ - \frac{y}{b} \tan θ = 1,$ it meets the axes at $L (a \cos θ, 0)$ and $M (0, - b \cot θ)$ .
Thus $(h, k)$ is given by $h = a \cos θ, k = - b \cot θ .$
Now eliminating $' θ'$ gives $\frac{a^{2}}{h^{2}} - \frac{b^{2}}{k^{2}} = 1.$

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