Question

If \( \theta \) is the angle made by the normal drawn to the curve \[ x = e^t \cos t, \quad y = e^t \sin t \] at the point \( (1, 0) \) with the X-axis, then \( \theta = \):

• A. \( \frac{\pi}{2} \)
• B. \( \frac{\pi}{4} \)
• C. \( \frac{3\pi}{2} \)
• D. \( \frac{3\pi}{4} \)

Hint:

Use parametric differentiation and reciprocal slope rule to find angle with X-axis.

Answer 1

The Correct Option is D

At point \( (1, 0) \), \( t = 0 \Rightarrow x = e^0 \cos 0 = 1, \, y = e^0 \sin 0 = 0 \) Find derivatives: \[ \frac{dx}{dt} = e^t(\cos t - \sin t), \quad \frac{dy}{dt} = e^t(\sin t + \cos t) \] Then slope of tangent = \( \frac{dy/dt}{dx/dt} \) Slope of normal = negative reciprocal. Angle \( \theta = \tan^{-1}(\text{slope of normal}) \Rightarrow \boxed{\frac{3\pi}{4}} \)