First slide

Parabola

Question

$y = - 2 x + 12 a$ is a normal to the parabola $y^{2} =$
$4 a x$ at the point whose distance from the directrix
of the parabola is

Moderate

A

$4 a$
B

$5 a$
C

$4 \sqrt{2} a$
D

$8 a$

Solution

Let $P$ be $(9 t^{2} / 4, 9 t / 2),$ then $N P = 9 t / 2 . M N = 9 t / 4$
equation of $M Q$ is $y = 9 t / 4$ which meets the parabola
$y^{2} = 9 x$ at the point $Q (9 t^{2} / 16, 9 t / 4), N$ is $(9 t^{2} / 4, 0)$
equation of $N Q$ is
$y = - \frac{9 \times 16}{4 \times 27 t} (x - \frac{9}{4} t^{2})$ which meets the tangent
$x = 0$ (at the vertex $A$ ) at the point for which
$y = 3 t$ so that $A T = 3 t$ and $\frac{A T}{N P} = \frac{3 t \times 2}{9 t} = \frac{2}{β} .$

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