Question

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

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

Answer 1

Hint: Differentiate the given curve equation and equate with the curve equation to find the points.
Given that:
Curve equation ${y^2} = 18x$
Ordinate increases twice the abscissa
So, $\dfrac{{dy}}{{dx}} = 2$ -- (1)
Differentiating the given parabola equation we get
$2ydy = 18dx \\\ \dfrac{{dy}}{{dx}} = \dfrac{{18}}{{2y}} \\\$ --- (2)
From equation 1 and 2, we have
$\dfrac{{18}}{{2y}} = 2 \\\ y = \dfrac{9}{2} \\\$
Substituting the value of $y$ obtained in the given curve equation:
$\Rightarrow {y^2} = 18x \\\ \Rightarrow \dfrac{{81}}{4} = 18x \\\ \Rightarrow x = \dfrac{9}{8} \\\$
Hence, the point is $\left( {\dfrac{9}{8},\dfrac{9}{2}} \right)$
Correct answer is option A.

Note:The following curve given in the question represents a parabola about x-axis. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus.