Solveeit Logo
Login

Question

Question: C is the centre of the hyperbola \(\frac{x^{2}}{a^{2}}\)– \(\frac{y^{2}}{b^{2}}\) = 1. The tangent a...

C is the centre of the hyperbola x2a2\frac{x^{2}}{a^{2}}– y2b2\frac{y^{2}}{b^{2}} = 1. The tangent at any point P on this hyperbola meets the straight line bx – ay = 0 and bx + ay = 0 in the points Q and R respectively then value of CQ . CR –

A

a2

B

b2

C

a2 + b2

D

None of these

Answer

a2 + b2

Explanation

Solution

P is (a sec q, b tan q)

tangent at P is xsec⁡θa\frac{x\sec\theta}{a} – ytan⁡θb\frac{y\tan\theta}{b}= 1

It meets bx – ay = 0 at Q

\ Q is (asec⁡θ−tan⁡θ,bsec⁡θ−tan⁡θ)\left( \frac{a}{\sec\theta - \tan\theta},\frac{b}{\sec\theta - \tan\theta} \right)

It meets bx + ay = 0 in R

\ R is (asec⁡θ+tan⁡θ,−bsec⁡θ+tan⁡θ)\left( \frac{a}{\sec\theta + \tan\theta},\frac{- b}{\sec\theta + \tan\theta} \right)

\ CQ . CR = a2+b2sec⁡θ−tan⁡θ\frac{\sqrt{a^{2} + b^{2}}}{\sec\theta - \tan\theta} . a2+b2sec⁡θ+tan⁡θ\frac{\sqrt{a^{2} + b^{2}}}{\sec\theta + \tan\theta}

= a2 + b2.