Question

If the tangent to the curve $x=a(8+sin\theta ),y=a(1+cos\theta )$ at $\theta =\frac{\pi }{3}$ makes an angle $\alpha$ with x-axis, then $\alpha$ is equal to

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

Answer 1

The correct option is B $\frac{2\pi }{3}$

$x=a(8+\sin \theta )$

$\frac{dx}{d\theta }=a\cos \theta$

$y=a(1+\cos \theta )$

$\frac{dy}{d\theta }=-a\sin \theta$

Now,

$\frac{dy}{d\theta }=-a\sin \theta$

$\frac{dx}{d\theta }=a\cos \theta$

$\frac{dy}{dx}=-\tan \theta$

$\frac{dy}{dx}(\mathrm{at}\theta =\frac{\pi }{3})=-\sqrt{3}=m$

Now,

$m=\tan \alpha$

$\Rightarrow \tan \alpha =-\sqrt{3}$

Hence,

$\alpha =\frac{2\pi }{3}$