Discussion Forum

If x= $e^{\theta}(\sin\theta-\cos\theta),y=e^{\theta}(\sin\theta+\cos\theta)$ then $\frac{dy}{dx}$at$\theta=\frac{\pi}{4}$   is 

Answer:

Explanation:

Given

   x= $e^{\theta}(\sin\theta-\cos\theta)$

and   $y=e^{\theta}(\sin\theta+\cos\theta)$ 

 $\Rightarrow$   $\frac{dx}{d\theta}=e^{\theta} (\sin \theta-\cos\theta+\cos\theta+\sin\theta)=2e^{\theta}\sin\theta$

 and  $\frac{dy}{d\theta}=e^{\theta} (\sin \theta+\cos\theta+\cos\theta-\sin\theta)=2e^{\theta}\cos \theta$

 $\therefore$     $\frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}=\frac{2e^{\theta}\cos \theta}{2e^{\theta}\sin \theta}=\cot \theta$

 $\Rightarrow$   $\left(\frac{dy}{dx}\right)_{\theta=\frac{\pi}{4}}=\cot \frac{\pi}{4}=1$