Physics Cup – TalTech 2021 – Pr. 2, Hint 2.

Probably the shortest solution can be obtained if we notice that the eccentricity vector $\vec \varepsilon = \hat r - \vec l \times \vec v$, where $\hat r$ denotes the unit vector pointing from the Sun to the comet, $\vec v$ — the velocity of the comet, and $\vec l$ — a constant vector perpendicular to the comet’s trajectory, can be expressed in complex plane as $\varepsilon = \mathrm e^{\mathrm i \varphi} + z$. Here $\varepsilon$ is the real-valued eccentricity, and $z$ is a complex number proportional in modulus to the speed.
NB! Using complex numbers here is a trick which leads you to a solution with just few lines. However, instead of using complex numbers you can also interpret the expression of the eccentricity vector geometrically, and solve a geometrical problem.

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