Question

If \( x = a \sec \theta, \, y = b \tan \theta \), then \( \frac{dy}{dx} = \):

• A. \( \frac{b}{a} \sec \theta \)
• B. \( \frac{b}{a} \csc \theta \)
• C. \( \frac{b}{a} \cot \theta \)
• D. \( \frac{b}{a} \)

Hint:

To find \( \frac{dy}{dx} \) when both \( x \) and \( y \) are functions of a third variable (like \( \theta \)), use: \[ \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} \]

Answer 1

The Correct Option is A

Given: \[ x = a \sec \theta \Rightarrow \frac{dx}{d\theta} = a \sec \theta \tan \theta \] \[ y = b \tan \theta \Rightarrow \frac{dy}{d\theta} = b \sec^2 \theta \] Now use the chain rule: \[ \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{b \sec^2 \theta}{a \sec \theta \tan \theta} = \frac{b}{a} \cdot \frac{\sec \theta}{\tan \theta} = \frac{b}{a} \cdot \frac{1}{\sin \theta} = \text{(incorrect)} \] Wait — let's correct this: \[ \frac{\sec \theta}{\tan \theta} = \frac{1/\cos \theta}{\sin \theta / \cos \theta} = \frac{1}{\sin \theta} = \csc \theta \] So: \[ \frac{dy}{dx} = \frac{b}{a} \csc \theta \] This matches Option (B), not (A). Correct Answer: (B) \( \frac{b}{a} \csc \theta \)
Correct Answer Correct Answer: (B) \( \frac{b}{a} \csc \theta \)