a
Initial momentum of the system is zero i.e. \(\mathrm{P}_{\mathrm{i}}=0\)

Let the velocity acquire by masses \(m_{1}\) and \(m_{2}\) just after they are released be \(v_{1}\) and \( v _ { 2 }\)

Final momentum of the system \(\mathrm{P}_{\mathrm{f}}=\mathrm{m}_{1} \mathrm{v}_{1}-\mathrm{m}_{2} \mathrm{v}_{2}\)

Using conservation of momentum: \(P_{i}=P_{f}\)

\(\therefore 0=m_{1} v_{1}-m_{2} v_{2}\)

\(\Rightarrow m_{1} v_{1}=m_{2} v_{2}\)         \(\left(v=\frac{d x}{d t}\right)\)

\(\Rightarrow m_{1} x_{1}=m_{2} x_{2}\)

\(\Rightarrow \frac{x_{1}}{x_{2}}=\frac{m_{2}}{m_{1}}\)