Central-force motion — UPSC IFoS 2019 Physics PYQ

A particle moves in an elliptical orbit in an inverse-square-law central force field. If the ratio of the maximum angular velocity to the minimum angular velocity of the particle in the orbit is n, then show that the eccentricity of the orbit is \varepsilon = \frac{\sqrt{n} - 1}{\sqrt{n} + 1}.