Question

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

• A. $\left(\frac{9}{8},\frac{9}{2}\right)$
• B. $(2,–4)$
• C. $\left(\frac{-9}{8},\frac{9}{2}\right)$
• D. $(2,4)$

Answer 1

The correct option is A

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

Explanation for the correct option:

Step 1. Find the point on parabola:

Given that, Ordinate increases at twice the rate of abscissa means

$dy=2dx$

$\Rightarrow$$\frac{dy}{dx}=2$

$\because y^2=18x$

Step 2. Differentiate it with respect to $x$:

$\Rightarrow$$2y\frac{dy}{dx}=18$

$\Rightarrow$$2y\times 2=18$

$\Rightarrow$ $y=\frac{9}{2}$

Now,

$\begin{array}{rcl}x& =& \frac{y^2}{18}\\ & =& \frac{81}{4\times 18}\\ & =& \frac{9}{8}\end{array}$

$\therefore$The required points are $\left(\frac{9}{8},\frac{9}{2}\right)$

Hence, Option ‘A’ is Correct.