Question

If $x=a(1-cos\theta ),y=a(\theta +sin\theta )$ then $\frac{dy}{dx}$ at $\theta =\frac{\pi }{2}$ is

• A. $1$
• B. $1/2$
• C. $0$
• D. $-1/2$

Answer 1

Answer: (A)

$1$

Given,

$x=a(1-\cos \theta )$

$\frac{dx}{d\theta }=a(0+\sin \theta )$

$y=a(\theta +\sin \theta )$

$\frac{dy}{d\theta }=a(1+\cos \theta )$

$\therefore \frac{dy}{dx}=\frac{a(1+\cos \theta )}{a\sin \theta }$

$\Rightarrow \frac{dy}{dx}=\frac{1+\cos \theta }{\sin \theta }$

$\Rightarrow {\frac{dy}{dx}}_{\theta =\frac{\pi }{2}}=\frac{1+\cos \frac{\pi }{2}}{\sin \frac{\pi }{2}}$

$\Rightarrow \frac{dy}{dx}=\frac{1-0}{1}=1$