Question

Consider the curve $x=a(\cos \theta +\theta \sin \theta )$ and $y=a(\sin \theta -\theta \cos \theta )$.

What is $\frac{dy}{dx}$ equal to.

• A. $\tan \theta$
• B. $\cot \theta$
• C. $\sin 2\theta$
• D. $\cos 2\theta$

Answer 1

The correct option is A $\tan \theta$

The given equation is:

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

$y=a(\sin \theta -\theta \cos \theta )$

Differentiating x and y w.r.t. to $\theta$ once we get,

$\Rightarrow \frac{dy}{d\theta }=a(\cos \theta -\cos \theta +\theta \sin \theta )$

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

$\Rightarrow \frac{dx}{d\theta }=a(\sin \theta -\sin \theta +\theta \cos \theta )$

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

Dividing the two equations we get,

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

$\Rightarrow \frac{dy}{dx}=\tan \theta$ .....Answer