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A conic section can be defined as the set of points whose distance to its focus is equal to the eccentricity times the distance to the corresponding directrix. Even in the case of two foci, the described set, applied on a single focus-directrix combination, is the whole conic section.
Note that (non-circular) ellipses and hyperbolas each have a pair of foci. An ellipse can be described as the set of points for which the sum of the distances to the foci is constant, while a hyperbola is the set of points for which the absolute value of the difference of the distances to the foci is constant.
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