Question

If \(x=a\cos^2\theta,\ y=b\sin^2\theta\), then the value of \(\dfrac{dy}{dx}\) is

• A. \(\dfrac{b}{a}\)
• B. \(-\dfrac{b}{a}\)
• C. \(\dfrac{b}{a}\sin 2\theta\)
• D. \(-\dfrac{b}{a}\tan^2\theta\)

Hint:

With parameters, divide derivatives: many trig factors cancel automatically.

Answer 1

The Correct Option is B

Idea. Use parametric differentiation: \(\dfrac{dy}{dx}=\dfrac{(dy/d\theta)}{(dx/d\theta)}\). Trig double-angle simplifies nicely.
Step 1. \(dy/d\theta=b\cdot 2\sin\theta\cos\theta=b\sin2\theta.\)
Step 2. \(dx/d\theta=a\cdot 2\cos\theta(-\sin\theta)=-a\sin2\theta.\)
Step 3. \(\dfrac{dy}{dx}=\dfrac{b\sin2\theta}{-a\sin2\theta}=-\dfrac{b}{a}\).