Question

If \(C\) is the centre of the hyperbola \[ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \] and the tangent drawn at any point \(P\) on the hyperbola meets the lines \[ bx-ay=0 \] and \[ bx+ay=0 \] at \(Q\) and \(R\) respectively, then \[ CQ\cdot CR= \]

• A. \(a^2-b^2\)
• B. \(a^2+b^2\)
• C. \[ \frac1{a^2}+\frac1{b^2} \]
• D. \[ \frac1{a^2}-\frac1{b^2} \]

Hint:

For hyperbola problems involving tangents, use the parametric point \[ (a\sec\theta,\; b\tan\theta) \] and the tangent \[ \frac{x\sec\theta}{a}-\frac{y\tan\theta}{b}=1. \] The identities \[ (\sec\theta+\tan\theta)(\sec\theta-\tan\theta)=1 \] often simplify the final expression dramatically.

Answer 1

The Correct Option is B