Question

If $x=a{\sec }^3\theta$ and $y=a{\tan }^3\theta$, then $\frac{dy}{dx}$ at $\theta =\frac{\pi }{3}$ is

• A. $\frac{\sqrt{3}}{2}$
• B. $\frac{\sqrt{5}}{2}$
• C. $\frac{\sqrt{2}}{3}$
• D. $\frac{\sqrt{4}}{3}$

Answer 1

The correct option is A $\frac{\sqrt{3}}{2}$

We have $x=a{\sec }^3\theta$ and $y=a{\tan }^3\theta$
Differentiate w. r to $\theta$
$\frac{dx}{d\theta }=3a{\sec }^2\theta \frac{d}{d\theta }\left(\sec \theta \right)=3a{\sec }^3\theta \tan \theta$
$\frac{dy}{d\theta }=3a{\tan }^2\theta \frac{d}{d\theta }\left(\tan \theta \right)=3a{\tan }^2\theta {\sec }^2\theta$
$\therefore \frac{dy}{dx}=\frac{\frac{dy}{d\theta }}{\frac{dx}{d\theta }}=\frac{3a{\tan }^2\theta {\sec }^2\theta }{3a{\sec }^3\theta \tan \theta }=\frac{\tan \theta }{\sec \theta }=\sin \theta$
${\left(\frac{dy}{dx}\right)}_{\theta =\frac{\pi }{3}}=\sin \frac{\pi }{3}=\frac{\sqrt{3}}{2}$