Hyperbola (redirected from Hiperbola)

hyperbola (hīpûr`bələ), plane curve consisting of all points such that the difference between the distances from any point on the curve to two fixed points (foci) is the same for all points. It is the conic section formed by a plane cutting both nappes of the cone; it thus has two parts, or branches. The center of a hyperbola is the point halfway between its foci. The principal axis is the straight line through the foci. The vertices are the intersection of this axis with the curve. The transverse axis is the line segment joining the two vertices. The latus rectum is the chord through either focus perpendicular to the principal axis. The asymptotes are lines, in the same plane, which the curve approaches as it approaches infinity. An equilateral, or rectangular, hyperbola is one whose asymptotes are perpendicular. A second hyperbola may be drawn whose asymptotes are identical with those of the given hyperbola and whose principal axis is a perpendicular line through the center; the two hyperbolas thus related are called conjugate.

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hyperbola

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 − y²/b² = 1. Hyperbolas have many important physical attributes that make them useful in the design of lenses and antennas.

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hyperbola

a conic section formed by a plane that cuts both bases of a cone; it consists of two branches asymptotic to two intersecting fixed lines and has two foci. Standard equation: x²/a² -- y²/b² = 1 where 2a is the distance between the two intersections with the x-axis and b = a&#221A(e² -- 1), where e is the eccentricity

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hyperbola [hī·pər·bə·lə]

(mathematics)

The plane curve obtained by intersecting a circular cone of two nappes with a plane parallel to the axis of the cone.

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Hyperbola 

the curve of intersection of a circular cone with a plane cutting both of its nappes (Figure 1). A hyperbola may also be defined as the geometric locus of the points M in a plane, such that the difference of their distances from two fixed points F₁ and F₂ (foci of the hyperbola) in that plane is constant. If a coordinate system xOy is selected such as

Figure 1

that represented in Figure 2 (OF₁ = OF₂ = c), then the equation of the hyperbola assumes the form

(2a = F₁M - F₂M and b = . A hyperbola is a curve of the second order. Consisting of two infinite branches K₁A₁K₁’ and K₂A₂K₂’, it is symmetrical with respect to the F₁F₂ and B₁B₂ axes. The point O is the center of the hyperbola and the center of its symmetry. The segments A₁A₂ = 2a and B₁B₂= 2b are called, respectively, the transverse and conjugate axes of the hyperbola. The number e = c/a > 1 is the eccentricity of the hyperbola. The straight lines D₁D₁’ and D₂D₂’, whose equations are x = -a/e and x = a/e, are the directrixes of the hyperbola. The ratio of the distance of a

Figure 2

point on the hyperbola from the nearest focus to the distance from the nearest directrix is constant and equal to the eccentricity. The points A₁ and A₂ of the hyperbola’s intersection with the Ox axis are called its vertices. The straight lines y = ± b/a (represented by dashed lines in Figure 2) are the asymptotes of the hyperbola. The graph of the inverse proportionality y = k/x is a hyperbola.