Step 1: Identify radius from equation. The given equation represents a circle with center at \( (0,0) \) and radius \( a \).
Step 2: Use area formula. \[ \text{Area} = \pi a^2 \]
Given:
\[ x \sqrt{1 + y} + y \sqrt{1 + x} + x = 0 \]for \( -1 < x < 1 \), prove that:
\[ \frac{dy}{dx} = -\frac{1}{(1+x)^2}. \]