# Dictionary Definition

hyperbola n : an open curve formed by a plane
that cuts the base of a right circular cone

# User Contributed Dictionary

## English

### Etymology

From (huperbolē)### Noun

hyperbola (plural hyperbolas or hyperbolae)- A conic section formed by the intersection of a cone with a plane that intersects the base of the cone and is not tangent to the cone.

#### Translations

geometric curve

#### See also

## Czech

### Noun

# Extensive Definition

distinguish hyperbole

In mathematics, a hyperbola
(Greek ,
"over-thrown") is a type of conic
section defined as the intersection between a right circular
conical
surface and a plane
which cuts through both halves of the cone.

It may also be defined as the locus of
points where the difference in the distance to two fixed points
(called the foci) is
constant. That fixed difference in distance is two times a where a
is the distance from the center of the hyperbola to the vertex of
the nearest branch of the hyperbola. a is also known as the
semi-major axis of the hyperbola. The foci lie on the transverse
axis and their midpoint is called the center.

For a simple geometric proof that the two
characterizations above are equivalent to each other, see Dandelin
spheres.

Algebraically, a hyperbola is a curve in the
Cartesian
plane defined by an equation of the form

- A x^2 + B xy + C y^2 + D x + E y + F = 0

The graph of two variables varying inversely on
the Cartesian coordinate plane is a hyperbola.

## Definitions

The first two were listed above:- The intersection between a right circular conical surface and a plane which cuts through both halves of the cone.

- The locus of points where the difference in the distance to two fixed points (called the foci) is constant.

- The locus of points for which the ratio of the distances to one focus and to a line (called the directrix) is a constant larger than 1. This constant is the eccentricity of the hyperbola.

A hyperbola comprises two disconnected curves called its arms or branches
which separate the foci. At large distances from the foci the
hyperbola begins to approximate two lines, known as asymptotes. The asymptotes
cross at the center of the hyperbola and have slope \pm \frac for
an East-West opening hyperbola or \pm \frac for a North-South
opening hyperbola.

A hyperbola has the property that a ray
originating at one of the foci is reflected
in such a way as to appear to have originated at the other focus.
Also, if rays are directed towards one of the foci from the
exterior of the hyperbola, they will be reflected towards the other
focus.

The simplest example of these are the hyperbolas

- y=\frac\,.

### Polar

East-west opening hyperbola:- r^2 =a\sec 2\theta \,

- r^2 =-a\sec 2\theta \,

- r^2 =a\csc 2\theta \,

- r^2 =-a\csc 2\theta \,

In all formulas the center is at the pole, and a
is the semi-major axis and semi-minor axis.

### Parametric

East-west opening hyperbola:- \begin

North-south opening hyperbola:

- \begin

In all formulae (h,k) are the center coordinates
of the hyperbola, a is the length of the semi-major axis, and b is
the length of the semi-minor axis.

## See also

## External links

hyperbola in Afrikaans: Hiperbool

hyperbola in Arabic: قطع زائد

hyperbola in Bulgarian: Хипербола

hyperbola in Catalan: Hipèrbola

hyperbola in Czech: Hyperbola

hyperbola in Danish: Hyperbel

hyperbola in German: Hyperbel (Mathematik)

hyperbola in Estonian: Hüperbool

hyperbola in Modern Greek (1453-): Υπερβολή
(γεωμετρία)

hyperbola in Spanish: Hipérbola

hyperbola in Esperanto: Hiperbolo

hyperbola in Persian: هذلولی

hyperbola in French: Hyperbole
(mathématiques)

hyperbola in Korean: 쌍곡선

hyperbola in Hindi: अति परवलय

hyperbola in Indonesian: Hiperbola
(matematika)

hyperbola in Italian: Iperbole (geometria)

hyperbola in Hebrew: היפרבולה

hyperbola in Georgian: ჰიპერბოლა

hyperbola in Lithuanian: Hiperbolė

hyperbola in Hungarian: Hiperbola

hyperbola in Dutch: Hyperbool (wiskunde)

hyperbola in Japanese: 双曲線

hyperbola in Norwegian: Hyperbel

hyperbola in Polish: Hiperbola
(matematyka)

hyperbola in Portuguese: Hipérbole

hyperbola in Romanian: Hiperbolă

hyperbola in Russian: Гипербола
(математика)

hyperbola in Slovak: Hyperbola

hyperbola in Slovenian: Hiperbola

hyperbola in Serbian: Хипербола

hyperbola in Finnish: Hyperbeli

hyperbola in Swedish: Hyperbel

hyperbola in Tamil: அதிபரவளைவு

hyperbola in Vietnamese: Hyperbol

hyperbola in Chinese: 双曲线