Question

The tangent at a point P on $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ cuts one of its directrices in Q. Then PQ subtends at the corresponding focus an angle of

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

Equation of the tangent at $P\left(\theta \right)=(asec\theta ,btan\theta )$ as the hyperbola is $\frac{xsec\theta }{a}-\frac{ytan\theta }{b}=1$
The tangent cuts the directrix $x=\frac{a}{e}atQ=(\frac{a}{e},\frac{b(sec\theta -e)}{etan\theta })$ and focus S = (ae, 0)
We get the product of slopes of $\overline{SP}and\overline{SQ}$ is –1