Question

A point on the parabola \(y^2 = \frac{36}{5}x\) at which the ordinate increases at thrice the rate of the abscissa is .....

• A. \((\frac{6}{5},\frac{1}{5})\)
• B. \((\frac{1}{5},\frac{6}{5})\)
• C. \((5,6)\)
• D. \((\frac{3}{5},\frac{9}{5})\)

Hint:

Differentiate with respect to time and set dy/dt equal to 3 dx/dt.

Answer 1

The Correct Option is B

Step 1: Use the parametric form.
The parabola $y^2 = 4ax$ with $4a = 36/5$ has $a = 9/5$. Take $x = at^2$, $y = 2at$.

Step 2: Rate condition.
$\dfrac{dy/dt}{dx/dt} = \dfrac{2a}{2at} = \dfrac{1}{t}$. Setting $1/t = 3$ gives $t = 1/3$.

Step 3: Coordinates.
$x = \dfrac{9}{5}\cdot\dfrac{1}{9} = \dfrac{1}{5}$ and $y = 2\cdot\dfrac{9}{5}\cdot\dfrac{1}{3} = \dfrac{6}{5}$.

Final Answer:
Option (B). \[ \boxed{\left(\frac{1}{5}, \frac{6}{5}\right)} \]