Question

If \( x = a(1 - \cos\theta) \), \( y = a(\theta - \sin\theta) \), then \( \frac{d^2y}{dx^2} = \)

• A. \( \frac{\cos^2\left(\frac{\theta}{2}\right)}{2a \csc\theta} \)
• B. \( \frac{\csc\theta}{2a \cos^2\left(\frac{\theta}{2}\right)} \)
• C. \( \frac{\cos\left(\frac{\theta}{2}\right)}{2a \sin\theta} \)
• D. \( \frac{\sin\left(\frac{\theta}{2}\right)}{2a \cos\theta} \)

Hint:

When differentiating parametric equations, always use the chain rule carefully to find the second derivative with respect to \( x \).

Answer 1

The Correct Option is B

Step 1: Differentiate \( y \) with respect to \( x \).
First, find \( \frac{dy}{d\theta} \) and \( \frac{dx}{d\theta} \), then use the chain rule to differentiate \( y \) with respect to \( x \).
Step 2: Differentiate again.
Next, compute \( \frac{d^2y}{dx^2} \) by differentiating \( \frac{dy}{dx} \) with respect to \( x \). The result is \( \frac{\csc\theta}{2a \cos^2\left(\frac{\theta}{2}\right)} \).
Step 3: Conclusion.
The correct answer is (B) \( \frac{\csc\theta}{2a \cos^2\left(\frac{\theta}{2}\right) \)}.