Hint 11

After the change in central mass, the small mass is still at radius $R$and has velocity $v=\sqrt{GM/R}$ in a tangential direction. However its total energy is now

$$E={1\over 2}mv^2 - {GM'm\over R}.$$

Use this to find the semimajor axis. Since the velocity is tangential to the orbit and greater than that of a body in a circular orbit of radius $R$ about a mass $M'$, $R$ is the pericentre distance of the new orbit. Hence determine the eccentricity. For $M'=3M/4$ you should get $a=2R$ and $\epsilon=0.5$.