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 :