I believe the center of mass for the earth moon system is something like 3000 km from the center of the earth (in a moonward direction). Just for reference, the radius of the earth is 6300 km.
The density of the earth is ~5.52g/cc, and halving the radius of the earth will increase the density by 8 which gives a density of 44.16 g/cc or 44,160 kg/m^3, which is about twice the density of osmium and a third as dense as the center of the sun.
In other words, the scenario you imagine is not practically possible in a system where one body is much larger than the other (if they're both about the same size, then it a dual planet system).
