Question

The equations \( x = a(\theta + \sin \theta) \) and \( y = a(1 - \cos \theta) \) represent the equation of a curve. If \( \theta \) changes at a constant rate \( k \), then the rate of change of the slope of the tangent to the curve at \( \theta = \frac{\pi}{3} \) is

• A. 2k
• B. \( \frac{k}{3} \)
• C. \( \frac{2k}{\sqrt{3}} \)
• D. \( \frac{2k}{3} \)

Hint:

When differentiating a quotient, use the quotient rule: \( \frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \).

Answer 1

The Correct Option is D