Question

If \( x = a \cos^2 t, y = b \sin^2 t \), then find \( \frac{dy}{dx} \).

Hint:

When finding \( \frac{dy}{dx} \) in parametric form, use the chain rule: \( \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \).

Answer 1

We are given that: \[ x = a \cos^2 t, y = b \sin^2 t \] To find \( \frac{dy}{dx} \), we use the chain rule: \[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \] Now, compute \( \frac{dy}{dt} \) and \( \frac{dx}{dt} \): 1. Differentiating \( y \) with respect to \( t \): \[ y = b \sin^2 t \] \[ \frac{dy}{dt} = b \cdot 2 \sin t \cdot \cos t = 2b \sin t \cos t \] 2. Differentiating \( x \) with respect to \( t \): \[ x = a \cos^2 t \] \[ \frac{dx}{dt} = a \cdot 2 \cos t (-\sin t) = -2a \cos t \sin t \] Thus, we have: \[ \frac{dy}{dx} = \frac{2b \sin t \cos t}{-2a \cos t \sin t} \] Simplifying: \[ \frac{dy}{dx} = -\frac{b}{a} \] Final Answer: \[ \boxed{\frac{dy}{dx} = -\frac{b}{a}} \]