Here is a step-by-step guide how to solve the problem. First we need to notice that the Kepler’s second law is none other than the conservation law for the angular momentum, and we can relate the rate at which a line segment connecting a satellite and the Earth’s centre sweeps out an area to the angular momentum of the satellite relative to the Earth. Next we need to write down the Runge-Lenz vector for the both of the satellites, and rescale each of them appropriately (for more details, see the Hint 4). As a result, we will be able to obtain expressions \hat z \times \vec v_i=k_i(\vec\varepsilon_i -\hat r_i), where the i=1,2 enumerates the satellites, cf. Hint 4. Finally, if we sketch the hodographs of the both satellites on the same plot, we can easily find the hodographs’ points corresponding to the largest difference between the velocities of the satellites, cf. Hint 4.
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