Question

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