The ordinates of the points P and Q on the parabola with focus (3, 0) and directrix are in the ratio 3 : 1. If is the point of intersection of the tangents to the parabola at P and Q, then is equal to _______. [2023]

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Solution

Q.
The ordinates of the points P and Q on the parabola with focus (3, 0) and directrix $x = - 3$ are in the ratio 3 : 1. If $R (α, β)$ is the point of intersection of the tangents to the parabola at P and Q, then $\frac{β^{2}}{α}$ is equal to _______. [2023]

Ans.
(16)

Parabola is $y^{2} = 12 x$

$Let Q (3 t^{2}, 6 t)$

$So, P (27 t^{2}, 18 t)$

$Now, R (α, β) = (a t_{1} t_{2}, a (t_{1} + t_{2}))$

$= (3 t \cdot 3 t, 3 (t + 3 t)) = (9 t^{2}, 12 t)$

$R (α, β) = (9 t^{2}, 12 t) \Rightarrow \frac{β^{2}}{α} = \frac{{(12 t)}^{2}}{9 t^{2}} = \frac{144}{9} = 16$

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