Question

The inclination of the tangent at $\theta =\frac{\pi }{3}$ on the curve $x=a(\theta +\sin \theta ),y=a(1+\cos \theta )$ is

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{6}$
• C. $\frac{2\pi }{3}$
• D. $\frac{5\pi }{6}$

Answer 1

The correct option is D $\frac{5\pi }{6}$

For the curve ,

$\frac{dy}{dx}=\frac{\frac{dy}{d\theta }}{\frac{dx}{d\theta }}=\frac{\frac{d\left(a\right(1+cos\theta \left)\right)}{d\theta }}{\frac{d\left(a\right(\theta +sin\theta \left)\right)}{d\theta }}=\frac{-asin\theta }{a+acos\theta }=\frac{-sin\theta }{1+cos\theta }$

So, If the inclination of tangent at $\theta =\frac{\pi }{3}$ be $y$ ,

Then, $\tan \left(y\right)=$ $\frac{-sin\left(\frac{\pi }{3}\right)}{1+cos\frac{\pi }{3}}=\frac{-1}{\sqrt{3}}$

Thus,$y=\pi -\frac{\pi }{6}=\frac{5\pi }{6}$