Question

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

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

Answer 1

The correct option is A $1$

We have,

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

$\frac{dx}{d\theta }={2e}^{\theta }sin\theta$

Also,

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

So,

$\frac{dy}{dx}=cot\theta$

and at $\theta =\pi /4$

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