Question

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

• A. $\cot \theta$
• B. $\cot \frac{\theta }{2}$
• C. $-\cot \frac{\theta }{2}$
• D. none of these

Answer 1

The correct option is C $-\cot \frac{\theta }{2}$

Given, $x=a(\theta -\sin \theta )$ and $y=a(1+\cos \theta )$.

Now $\frac{dx}{d\theta }=a(1-\cos \theta )$.....(1) and $\frac{dy}{d\theta }=-a\sin \theta$......(2).

From (1) and (2) we get,

$\frac{dy}{dx}=\frac{\frac{dy}{d\theta }}{\frac{dx}{d\theta }}=\frac{-a\sin \theta }{a(1-\cos \theta )}=\frac{-2\sin \frac{\theta }{2}.\cos \frac{\theta }{2}}{2{\sin }^2\frac{\theta }{2}}=-\cot \frac{\theta }{2}$.