Let a satellite of mass $M$ revolves around earth with a velocity $V$. Consider ideal conditions. No sun’s and other planets gravitational force. What happens to satellite when its mass is suddenly reduced to half of its original mass i.e $M/2$?
No change will happen, as gravitational force will become half as well, so acceleration (=force/mass) will remain same. Note that the speed of a particle in a gravitational orbit does not depend on its mass. Therefore, the satellite will continue to travel in the same orbit.
For the same reason, in absence of air drag, all objects at sea level fall with same acceleration.
However, if you want to conserve momentum after changing mass (not conserve the velocity), the trajectory will change. Since you have written that satellite was travelling with speed $v$ (always), the the orbit must be circular. Then, if momentum is conserved (speed will be doubled just after mass is halved), the satellite will fly away in a parabolic orbit.
You don't suddenly "lose mass"... my scales tell me this every day.
In order to answer your question, we have to consider "where did the mass go?". This is because other conservation laws (momentum, angular momentum) come into play. If you say "mass is halved but my angular momentum is unchanged" then your orbital velocity doubled, and your satellite will start to describe a different path. If the mass is lost symmetrically (a little bit in each direction) then the remaining satellite will carry on as before (which you can see by considering the satellite to be made up of two equal halves; if these halves were orbiting together, connected by a piece of string, they will still orbit together if the string is cut).