Differential Equations

The area enclosed by the closed curve $\mathrm{C}$ given by the differential equation $\frac{d y}{d x}+\frac{x+a}{y-2}=0, y(1)=0$ is $4 \pi$. Let $P$ and $Q$ be the points of intersection of the curve $\mathrm{C}$ and the $y$-axis. If normals at $P$ and $Q$ on the curve $\mathrm{C}$ intersect $x$-axis at points $R$ and $S$ respectively, then the length of the line segment $R S$ is :