# hyperbola

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hyperbola
/huy perr"beuh leuh/, n. Geom.
the set of points in a plane whose distances to two fixed points in the plane have a constant difference; a curve consisting of two distinct and similar branches, formed by the intersection of a plane with a right circular cone when the plane makes a greater angle with the base than does the generator of the cone. Equation: x2/a2 - y2/b2 = ±1. See diag. under conic section.
[1660-70; < NL < Gk hyperbolé the geometrical term, lit., excess. See HYPERBOLE]

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Curve with two separate branches, one of the conic sections.

In Euclidean geometry, the intersection of a double right circular cone and a plane at an angle that is less than the cone's generating angle (the angle its sides make with its central axis) forms the hyperbola's two branches (one on each nappe, or single cone). In analytic geometry, the standard equation of a hyperbola is x2/a2/n-/ny2/b2/n=/n1. Hyperbolas have many important physical attributes that make them useful in the design of lenses and antennas.

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two-branched open curve, a conic section, produced by the intersection of a circular cone and a plane that cuts both nappes (see cone) of the cone. As a plane curve it may be defined as the path (locus) of a point moving so that the ratio of the distance from a fixed point (the focus) to the distance from a fixed line (the directrix) is a constant greater than one. The hyperbola, however, because of its symmetry, has two foci. Another definition is that of a point moving so that the difference of its distances from two fixed points, or foci, is a constant. A degenerate hyperbola (two intersecting lines) is formed by the intersection of a circular cone and a plane that cuts both nappes of the cone through the apex.

A line drawn through the foci and prolonged beyond is the transverse axis of the hyperbola; perpendicular to that axis, and intersecting it at the geometric centre of the hyperbola, a point midway between the two foci, lies the conjugate axis. The hyperbola is symmetrical with respect to both axes.

Two straight lines, the asymptotes of the curve, pass through the geometric centre. The hyperbola does not intersect the asymptotes, but its distance from them becomes arbitrarily small at great distances from the centre. The hyperbola when revolved about either axis forms a hyperboloid (q.v.).

For a hyperbola that has its centre at the origin of a Cartesian coordinate system and has its transverse axis lying on the x axis, the coordinates of its points satisfy the equation x2/a2 - y2/b2 = 1, in which a and b are constants.

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Universalium. 2010.

### Look at other dictionaries:

• Hyperbola — Hy*per bo*la, n. [Gr. ?, prop., an overshooting, excess, i. e., of the angle which the cutting plane makes with the base. See {Hyperbole}.] (Geom.) A curve formed by a section of a cone, when the cutting plane makes a greater angle with the base… …   The Collaborative International Dictionary of English

• hyperbola — 1660s, from Latinized form of Gk. hyperbole extravagance, lit. a throwing beyond (see HYPERBOLE (Cf. hyperbole)). Perhaps so called because the inclination of the plane to the base of the cone exceeds that of the side of the cone …   Etymology dictionary

• hyperbola — ► NOUN (pl. hyperbolas or hyperbolae) ▪ a symmetrical open curve formed by the intersection of a cone with a plane at a smaller angle with its axis than the side of the cone. ORIGIN Latin, from Greek huperbol (see HYPERBOLE(Cf. ↑hyperbolically)) …   English terms dictionary

• hyperbola — [hī pʉr′bə lə] n. pl. hyperbolas or hyperbolae [hī pʉr′bəlē] [ModL < Gr hyperbolē, a throwing beyond, excess < hyperballein, to throw beyond < hyper (see HYPER ) + ballein, to throw (see BALL1)] Geom. the path of a point that moves so… …   English World dictionary