Question

If the parametric equation of a curve given by $x=e^t\cos t,y=e^t\sin t$ , then the tangent to the curve at the point $=\frac{\pi }{4}$ makes with axis of $x$ , the angle is

• A. $0$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{3}$
• D. $\frac{\pi }{2}$

Answer 1

Answer: (D)

We have, $x=e^t\cos t$ and $y=e^t\sin t$
Therefore, $\frac{dx}{dt}=e^t(\cos t-\sin t)$ and $\frac{dy}{dt}=e^t(\sin t+\cos t)$
$\therefore$ $\frac{dy}{dx}=\frac{dy/dt}{dx/dt}=\frac{\sin t+\cos t}{\cos t-\sin t}$
$\Rightarrow$ ${\left(\frac{dy}{dx}\right)}_{t=\pi /4}=\infty =\tan \frac{\pi }{2}$
So, tangent at $t=\pi /4$ makes with axis of $x$ , the angle is $\pi /2$ .