The portion of the tangent intercepted between the point of contact and the directrix of the parabola $ y^2 = 4ax$ subtends at the focus an angle of

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Solution

The correct option is D
If $P = (a t^{2}, 2 a t)$ then the equation of tangent at t is $t y = x + a t^{2} . . . . . . . (1)$
Directrix is $x + a = 0 . . . . . . . . . . . . (2)$
By solving $Q = [- a, \frac{a (t^{2} - 1)}{t}], S = (a, 0)$
Slope of $S P = \frac{2 a t}{a (t^{2} - 1)} = \frac{2 t}{t^{2} - 1}$ ,Slope of $S Q = \frac{\frac{a (t^{2} - 1)}{t}}{- 2 a} = \frac{- (t^{2} - 1)}{2 t}$
Product of slopes = –1. $∴ ∠ P S Q = 90^{\circ}$

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