Let PQ be a focal chord of the parabola $y^{2} = 36 x$ of length 100, making an acute angle with the positive x-axis. Let the ordinate of P be positive and M be the point on the line segment PQ such that $PM : MQ = 3 : 1$ . Then which of the following points does NOT lie on the line passing through M and perpendicular to the line PQ?

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(-6, 45)

(6, 29)

(-3, 43)

(3, 33)

answer is D.

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Detailed Solution

Let $P = ({at}^{2}, 2 at) = (9 t^{2}, 18 t)$
$T h e n (\frac{9}{t^{2}}, \frac{- 18}{t}) PQ = 100 \Rightarrow 9 {(t + \frac{1}{t})}^{2} = 100$
$\Rightarrow t + \frac{1}{t} = \frac{10}{3}$
$\Rightarrow t = 3 (or) \frac{1}{3} P = (81, 54) or (1, 6) ∵ Focus S = (9, 0)$
$∵ SP$ makes a cute angle with $+ ve x - axis, P = (81, 54)$ $\Rightarrow Q = (1, - 6); M = (21, 9)$
Equation of the line through M and perpendicular to PQ is $4 x + 3 y = 111$
$(- 3, 43)$ does not lie on above line.

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