Question

If $x=3\sin t,y=3\cos t,$ then $\frac{dy}{dx}$ at $t=\frac{\pi }{3}$ is equal to

• A. $3$
• B. $0$
• C. $-\sqrt{3}$
• D. $1$

Answer 1

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

We have,
$x=3\sin t$
$\Rightarrow \frac{dx}{dt}=3\cos t$
and, $y=3\cos t$
$\Rightarrow \frac{dy}{dt}=-3\sin t$
$\therefore \frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$
$\Rightarrow \frac{dy}{dx}=\frac{-3\sin t}{3\cos t}=-\tan t$
At $t=\frac{\pi }{3},$
$\frac{dy}{dx}=-\sqrt{3}$