The tangent at a point P on the hyperbola x^2/a^2 – y^2/b^2 = 1 meets one of its directrices in F.

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

(A) π/4

(B) π/2

(C) 3π/4

(D) π

Sol. Let directrix be x = a/e and focus be S(ae, 0). Let P( asecθ , btanθ) be any point on the curve,

Equation of tangent at ‘P’ is $\large \frac{x sec\theta}{a} – \frac{y tan\theta}{b} = 1 $

Let ‘F’ be the intersection point of tangent of directrix, then

$\large F \equiv (a/e , \frac{b(sec\theta – e)}{e tan\theta})$

$\large m_{SF} = \frac{b(sec\theta – e)}{-a tan\theta (e^2 -1)} $

$\large m_{PS} = \frac{b tan\theta}{a(sec\theta -e)} $

⇒ m_SF . m_PS = –1

Hence (B) is the correct answer.