I shall assume that the sun is at rest, so before the half mass disappear, the orbit of earth is an ellipse.
Answer
Earth will escape from the solar system (It no longer in a)
IF you remember the Virial Theorem in order for a planet to have an orbit the average potential energy and average kinetic energy of a planet must have a relationship of 2Kavg=−Uavg
Where U is a gravitational potential energy, which is given by Uavg=−GMsunmplanetRplanet
If you reduce the mass of the sun by a half, potential energy will be half too.
Thus, now, 2Kavg>−12Uavg=−Uavg,new
In such case, the velocity of the planet is the escape velocity of the new sun. The planet will fly away from the sun to somewhere else and becomes a massive interstellar junk.
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