Question

The tangent at a point $P$ on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ meets one of the directrix in $F$. If $PF$ subtends an angle $\theta$ at the corresponding focus, the $\theta$ equals :

• A. $\frac{\pi }{4}$
• B. $\frac{\pi }{2}$
• C. $\frac{3\pi }{4}$
• D. $\pi$

Answer 1

The correct option is B $\frac{\pi }{2}$

Let the directrix be $x=\frac{a}{e}$ and focus be $S(ae,0)$.
Let $P(a\sec \theta ,b\tan \theta )$ be any point on the curve.
Equation of tangent at $P$ is $\frac{x\sec \theta }{a}-\frac{y\tan \theta }{b}=1$
Let $F$ be the intersection of tangent and the directrix, then
$F=(\frac{a}{e},\frac{b\left(\sec \theta -e\right)}{e\tan \theta })$
$\Rightarrow m_{SF}=\frac{b\left(\sec \theta -e\right)}{-e\tan \theta (a^2-1)},m_{PS}=\frac{b\tan \theta }{a(\sec \theta -e)}\Rightarrow m_{SF}\times m_{PS}=-1$