First slide

Parabola

Question

A point on the parabola $y^{2} = 18 x$ at which the ordinate increases at twice the rate of the abscissa is

Moderate

A

$(- \frac{9}{8}, \frac{9}{2})$
B

$(2, - 4)$
C

$(2, 4)$
D

$(\frac{9}{8}, \frac{9}{2})$

Solution

$y^{2} = 18 x \Rightarrow 2 y \frac{d y}{d x} = 18$
$\Rightarrow \frac{d y}{d x} = \frac{9}{y}$
We are given $\frac{d y}{d x} = 2$
so $\frac{9}{y} = 2 \Rightarrow y = \frac{9}{2}$
and $x = \frac{1}{18} y^{2} \Rightarrow x = \frac{9}{8}$
and the required point is $(\frac{9}{8}, \frac{9}{2})$

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