Hyperbola

This article isn't about hyperbole, a figure of speech. For that article, see hyperbole.

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A hyperbola is a type of conic section.

• Geometrically, it is defined as the intersection between a cone and a plane which cuts through both halves of the cone.

• Analytically, it is defined as the set of all points for which the difference in the distance to two fixed points (called the foci) is constant.

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

• It can also be defined as 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. These foci lie on the transverse axis and their midpoint is called the center.

A hyperbola comprises two disconnected curves called its arms which separate the foci. At large distances from the foci the hyperbola begins to approximate two lines, known as asymptotes.

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.

A special case of the hyperbola is the equilateral or rectangular hyperbola, in which the asymptotes intersect at right angles. The rectangular hyperbola with the co-ordinate axes as its asymptotes is given by the equation xy=c, where c is a constant.

Just as the sine and cosine functions give a parametric equation for the ellipse, so the hyperbolic sine and hyperbolic cosine give a parametric equation for the hyperbola.

A body that has sufficient energy to escape the gravitational field of a massive body moves in a hyperbolic trajectory with the massive body at one of the foci.

Equations (Cartesian):

(center (h, k) )

<math>\frac{\left( x-h \right)^2}{a^2} - \frac{\left( y-k \right)^2}{b^2} = 1<math>

<math>\frac{\left( y-k \right)^2}{a^2} - \frac{\left( x-h \right)^2}{b^2} = 1<math>

<math>(x-h)(y-k) = c \,<math>

Equations (polar):

<math>r^2 =\ \ \, a\,\sec 2t<math>

<math>r^2 = -a\,\sec 2t<math>

<math>r^2 =\ \ \, a\,\csc 2t<math>

<math>r^2 = -a\,\csc 2t<math>

Equations (parametric):

<math>x = a\,\cosh \theta;\; y = b\,\sinh \theta<math>

<math>x = a\,\tan \theta;\ \ y = b\,\sec \theta<math>

See also

• Ellipse
• parabola
• conic section
• Dandelin spheres
• hyperbole