Question

If $y=a{\sin }^3\theta$ and $x=a{\cos }^3\theta$, then at $\theta =\frac{\pi }{3},\frac{dy}{dx}$ is equal to:

• A. $\frac{1}{\sqrt{3}}$
• B. $-\sqrt{3}$
• C. $\frac{-1}{\sqrt{3}}$
• D. $\sqrt{3}$

Answer 1

The correct option is B $-\sqrt{3}$

Given, $y=a{\sin }^3\theta$ and $x=a{\cos }^3\theta$

On differentiating with respect to $\theta$, we get

$\frac{dy}{d\theta }=3a{\sin }^2\theta \cos \theta$

and $\frac{dx}{d\theta }=-3a{\cos }^2\theta \sin \theta$

$\therefore \frac{dy}{dx}=\frac{\frac{dy}{d\theta }}{\frac{dx}{d\theta }}=-\frac{3a{\sin }^2\theta \cos \theta }{3a{\cos }^2\theta \sin \theta }$

$=-\frac{\sin \theta }{\cos \theta }=-\tan \theta$

At $\theta =\frac{\pi }{3},\frac{dy}{dx}=-\tan \frac{\pi }{3}=-\sqrt{3}$