Question

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

• A. $(2,6)$
• B. $(2,-6)$
• C. $(\frac{9}{8},-\frac{9}{2})$
• D. $(\frac{9}{8},\frac{9}{2})$

Answer 1

The correct option is D $(\frac{9}{8},\frac{9}{2})$

We have $y^2=18x...\left(1\right)$
$\therefore 2y\frac{dy}{dx}=18\text{or}\frac{dy}{dx}=\frac{9}{y}$
Given that $\frac{dy}{dx}=2$
$\Rightarrow \frac{9}{y}=2\Rightarrow y=\frac{9}{2}$

Putting in $\left(1\right),$ we get
$\frac{81}{4}=18x\Rightarrow x=\frac{9}{8}$

Hence, the point is $(\frac{9}{8},\frac{9}{2}).$