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: Understanding the Concept:
"Ordinate increases at thrice the rate of abscissa" means \(\dfrac{dy}{dt} = 3\dfrac{dx}{dt}\), so \(\dfrac{dy}{dx} = 3\).

Step 2: Differentiate the curve.
\[ y^2 = \frac{36}{5}x \Rightarrow 2y\frac{dy}{dx} = \frac{36}{5} \Rightarrow \frac{dy}{dx} = \frac{18}{5y} \]

Step 3: Apply the condition.
\[ \frac{18}{5y} = 3 \Rightarrow y = \frac{6}{5} \]
Find \(x\) from the curve: \(x = \dfrac{5}{36}y^2 = \dfrac{5}{36}\cdot\dfrac{36}{25} = \dfrac{1}{5}\).

Step 4: Check the options.
The point is \(\left(\tfrac{1}{5}, \tfrac{6}{5}\right)\), which is option (B). Option (A) has the coordinates swapped and does not satisfy the curve.

Final Answer:
The required point is \(\left(\dfrac{1}{5}, \dfrac{6}{5}\right)\). \[ \boxed{\left(\frac{1}{5}, \frac{6}{5}\right)} \]