Question

If $x=e^{\theta }\sin \theta ,y=e^{\theta }\cos \theta$ where $\theta$ is a parameter, then $\frac{dy}{dx}$ at $(1,1)$ is equal to

• A. $0$
• B. $-\frac{1}{2}$
• C. $\frac{1}{2}$
• D. $-\frac{1}{4}$

Answer 1

Answer: (A)

$x=e^{\theta }\sin \theta =1$
$y=e^{\theta }\cos \theta =1,\frac{x}{y}=\tan \theta =1$
$\Rightarrow \theta =\frac{\pi }{4}$
$\frac{dy}{dx}=\left|\frac{dy/d\theta }{dx/d\theta }\right|=\frac{-e^{\theta }\sin \theta +\cos \theta \cdot e^{\theta }}{e^{\theta }\cos \theta +\sin \theta e^{\theta }}=\frac{\cos \theta -\sin \theta }{\cos \theta +\sin \theta }$
$=\tan \left(\frac{\pi }{4}-\theta \right)=0$