# hyperbola

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Related to hyperbolae: hyperbole

## 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**conic section**

or

**conic**

, curve formed by the intersection of a plane and a right circular cone (conical surface). The ordinary conic sections are the circle, the ellipse, the parabola, and the hyperbola.

**.....**Click the link for more information. formed by a plane cutting both nappes of the cone

**cone**

or

**conical surface,**

in mathematics, surface generated by a moving line (the generator) that passes through a given fixed point (the vertex) and continually intersects a given fixed curve (the directrix).

**.....**Click the link for more information. ; 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.

## hyperbola

(hÿ-**per**-bŏ-lă) A type of conic section that has an eccentricity greater than one. See also orbit.

## Hyperbola

## 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 F_{1} and F_{2} (foci of the hyperbola) in that plane is constant. If a coordinate system *xOy* is selected such as

that represented in Figure 2 *(OF _{1} = OF_{2} = c)*, then the equation of the hyperbola assumes the form

(2a = F_{1}M - F_{2}M and b = . A hyperbola is a curve of the second order. Consisting of two infinite branches *K*_{1}*A*_{1}*K*_{1}’ and *K*_{2}*A*_{2}*K*_{2}’, it is symmetrical with respect to the *F*_{1}*F*_{2} and *B*_{1}*B*_{2} axes. The point *O* is the center of the hyperbola and the center of its symmetry. The segments *A*_{1}*A*_{2} = 2a and *B*_{1}*B*_{2}= 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 *D*_{1}*D*_{1}’ and *D*_{2}*D*_{2}’, whose equations are x = *-a/e* and *x* = *a/e*, are the directrixes of the hyperbola. The ratio of the distance of a

point on the hyperbola from the nearest focus to the distance from the nearest directrix is constant and equal to the eccentricity. The points *A*_{1} and *A*_{2} 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.

## hyperbola

[hī·pər·bə·lə]## hyperbola

*x*

^{2}/

*a*

^{2}--

*y*

^{2}/

*b*

^{2}= 1 where 2

*a*is the distance between the two intersections with the

*x*-axis and

*b*=

*a*ÝA(

*e*

^{2}-- 1), where

*e*is the eccentricity