If \(x = a\sec^3\theta, y = a\tan^3\theta\), then \(\displaystyle\frac{dy}{dx}\) at \(\theta = \displaystyle\frac{\pi}{3}\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{\sqrt{3}}{2}\)
For a curve in parametric form, \(\frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta}\). By the chain rule:
\( \displaystyle \frac{dx}{d\theta} = 3a\sec^2\theta \cdot \sec\theta\tan\theta = 3a\sec^3\theta\tan\theta \)
\( \displaystyle \frac{dy}{d\theta} = 3a\tan^2\theta \cdot \sec^2\theta \)
Dividing:
\( \displaystyle \frac{dy}{dx} = \frac{3a\tan^2\theta\sec^2\theta}{3a\sec^3\theta\tan\theta} = \frac{\tan\theta}{\sec\theta} = \sin\theta \)
At \(\theta = \frac{\pi}{3}\):
\( \displaystyle \frac{dy}{dx} = \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2} \)
NCERT Exemplar:
Continuity and Differentiability
