Solution

Since total momentum of the system is zero, the center of mass (C.M.) remains stationary. The C.M. is on the line segment joining the two particles so that it is twice as far from the particle with mass $m$ as it is from the particle with mass $2m$. Thus, the trajectory of the particle with mass $m$ is obtained by reflecting the trajectory of the particle with mass $2m$ across the C.M. and enlarging it by a factor of 2.