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 $ .