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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