For any isolated system of two bodies revolving around each other by virtue of the gravitational attraction that each one exerts on the other, the general motion is best described by using a frame of reference attached to their common Center of Mass (CM). The reason is that their motion is in fact around their CM as we shall see.
For an isolated system the momentum remains constant so that the CM is either moving along a straight line or is at rest.
For an Earth's satellite we can always take the motion of the satellite relative to Earth using a geocentric frame of reference.
The reason is that:
                                         the mass of the satellite being insignificant compared to Earth's
                                         mass, the revolving satellite doesn't affect Earth at all so
                                         that the CM of Earth-satellite system is still the center of the Earth.
                                         Hence we use the center of the Earth as the origin of a rectangular
                                         coordinates system.
                                       
In this article we use Maple powerful animation routines to study the motion of two bodies having comparable masses revolving about each other by showing:
1- their combined motion as seen from their common Center of  Mass,
2- their relative motion as if one of them is fixed and the other one is moving.
In this last instance the frame of reference is attached to the the body that is supposed to be at rest.