Hyperbola

Article By:

Blumenthal, Leonard M. Formerly, Department of Mathematics, University of Missouri, Columbia, Missouri.

Last reviewed:August 2020

DOI:https://doi.org/10.1036/1097-8542.332000

A curve cut from a cone of revolution by a plane that intersects both nappes of the cone and does not contain the apex (Fig. 1). In analytic geometry it is shown, as shown in Fig. 2, that a hyperbola is the locus of points P in a plane, such that PF = ε · PD, where PF and PD denote the distances of P from a fixed point F (focus) and a fixed line (directrix) of the plane, respectively, and ε is a constant, greater than 1. It is also the locus of points P, the difference of whose distances from two fixed points F, F′ (foci) PF − PF′ is a constant 2a that is less than the distance 2c between the foci. The curve is symmetric to the line g(F, F′) determined by F, F′ and to O, their midpoint. It consists of two branches that are images of each other in the line g through O, perpendicular to g(F, F′). There are two lines through O, making equal angles with g(F, F′), and to each of which points on each branch get indefinitely close; that is, if point P traverses either branch of the hyperbola, its distance from these lines approaches zero.

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