Family of Circles Governed by First-Order Differential Equation

Separating variables in $\frac{dy}{dx} = \frac{\sqrt{1 - y^2}}{y}$:
$\frac{y}{\sqrt{1 - y^2}}\,dy = dx$

Integrating both sides:
$\int \frac{y}{\sqrt{1 - y^2}}\,dy = \int dx$

Let $u = 1 - y^2$, then $du = -2y\,dy \implies y\,dy = -\frac{du}{2}$:
$-\frac{1}{2}\int u^{-1/2}\,du = x + C_1 \implies -\sqrt{1 - y^2} = x - c$
$\sqrt{1 - y^2} = -(x - c)$

Squaring both sides:
$1 - y^2 = (x - c)^2 \implies (x - c)^2 + y^2 = 1$

This represents a family of circles with fixed radius $r = 1$ and variable centres $(c, 0)$ along the $x$-axis.

Correct Answer: C