Question

If \( x = a \sin \theta \) and \( y = b \cos \theta \), then \( \frac{d^2y}{dx^2} \) is:

• A. \( \frac{a}{b^2} \sec^2 \theta \)
• B. \( \frac{-b}{a^2} \sec^3 \theta \)
• C. \( \frac{-a}{b^2} \sec^3 \theta \)
• D. \( \frac{b}{a^2} \sec^3 \theta \)

Hint:

Use the chain rule for differentiating parametric equations to find higher derivatives.

Answer 1

The Correct Option is B

Step 1: Differentiate \( y \) with respect to \( x \).
We start by finding \( \frac{dy}{dx} \) and then differentiate again to find \( \frac{d^2y}{dx^2} \). Using the chain rule and applying the given relations, we get the desired result \( \frac{-b}{a^2} \sec^3 \theta \).
Thus, the correct answer is (2).