Question

If \( x = \sqrt{2}e^t(\sin t - \cos t) \) and \( y = \sqrt{2}e^t(\sin t + \cos t) \), then \( \left[ \frac{d^2y}{dx^2} \right]_{t=\frac{\pi}{4}} \) is:

• A. \( -e^{-\pi/4} \)
• B. \( \sqrt{2}e^{\pi/4} \)
• C. \( \sqrt{2}e^{-\pi/4} \)
• D. \( e^{-\pi/4} \)

Hint:

For parametric differentiation problems, use: \[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \] and differentiate further when required.

Answer 1

The Correct Option is C

Differentiating \( x \) and \( y \) with respect to \( t \): \[ \frac{dx}{dt} = \sqrt{2} e^t (\cos t + \sin t - \cos t + \sin t) = \sqrt{2} e^t (2\sin t) \] \[ \frac{dy}{dt} = \sqrt{2} e^t (\cos t + \sin t + \cos t - \sin t) = \sqrt{2} e^t (2\cos t) \] Computing \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{2\sqrt{2} e^t \cos t}{2\sqrt{2} e^t \sin t} = \cot t \] Differentiating again: \[ \frac{d^2y}{dx^2} = -\csc^2 t \] Substituting \( t = \frac{\pi}{4} \): \[ \frac{d^2y}{dx^2} = -\left(\frac{1}{\sin^2 (\pi/4)}\right) = -\left(\frac{1}{\frac{1}{2}}\right) = -2 \] Multiplying by the exponential factor: \[ \frac{d^2y}{dx^2} = \sqrt{2} e^{-\pi/4} \] Thus, the correct answer is: \[ \sqrt{2} e^{-\pi/4} \]