Let us start with the Runge-Lenz vector as written in the IPhO formula sheet (formula XII-9) in which case its length equals to the ellipticity of the orbit. If we multiply it by k\equiv GMm/L, where m is the satellite’s mass and L – its angular momentum with respect to the centre of the Earth, we obtain \hat z \times \vec v=k(\vec\varepsilon -\hat r), where \hat r denotes a unit vector pointing from the centre of the Earth in the direction of the satellite, and \hat z – a unit vector perpendicular to the orbital plane. Note that k is constant, hence the last equality means that the satellite’s hodograph is a circle (in other words, if we fix the the starting point of its velocity vector, the ending point draws a circle). This is valid for the both satellites. Now, the maximal relative speed can be easily found from the hodographs of the two satellites.
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