First slide

Parabola

Question

$P N$ is an ordinate of the parabola $y^{2} = 9 x$ . $A$ straight
line is drawn through the mid-point $M$ of $P N$ parallel
to the axis of the parabola meeting the parabola
at $Q$ . $N Q$ meets the tangent at the vertex $A$ , at a
point $T$ , then $A T / N P =$

Moderate

A

3/2
B

4/3
C

2/3
D

3/4

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 $A T = 3 t_{1}$ and $\frac{A T}{N P} = \frac{3 t \times 2}{9 t} = \frac{2}{3} .$

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