Question

Let the equation of a curve be $x=a(\theta +\sin \theta )$, $y=a(1-\cos \theta )$. If $\theta$ changes at a constant rate $k$ then the rate of change of slope of the tangent to the curve at $\theta =\frac{\pi }{2}$ is

• A. $\frac{2k}{\sqrt{3}}$
• B. $\frac{k}{\sqrt{3}}$
• C. $k$
• D. none of these

Answer 1

The correct option is C $k$

As $\theta$ change with constant rate $\Rightarrow \frac{d\theta }{dt}=k$

$x=a(\theta +sin\theta )$
$\frac{dx}{d\theta }=a(1+cos\theta )$
$y=a(1-cos\theta )$
$\frac{dy}{d\theta }=asin\theta$
Then, $\frac{dy}{dx}=\frac{asin\theta }{a(1+cos\theta )}$

Rate of change of slope $=\frac{d}{dt}.\frac{dy}{dx}=\frac{1}{1+\cos \theta }.\frac{d\theta }{dt}=k$ , at $\theta =\frac{\pi }{2}$