# Hyperbola

By the end of this lesson, you will be able to

- define hyperbola.
- identify different parts of an hyperbola.
- interpret different parts of an hyperbola.
- draw an hyperbola.

## Hyperbola

A hyperbola is a set of two pieces of smooth curves formed by the set of all points such that the difference of two absolute distances from two fixed points to every point on each curve is constant.

Different parts of hyperbola

Each fixed point is called the focus of the hyperbola. The plural of focus is foci. The midpoint of line segment connected to two foci is the **center** of the hyperbola.

## Focus

A focus or foci of a hyperbola are two special points such that the difference of two absolute distances from two special points to every point on the hyperbola is constant.

## Center

The midpoint of line segment joining two foci is the center of hyperbola. Center can be expressed in terms of axes. It is also the point of intersection of major axis and minor axis. Again, the two axes of a hyperbola meet at right angle at the center.

## Major Axis

A major axis is a a href="lines-and-angles">line passes through the foci of the hyperbola. It is also the axis of symmetry. The length of the major axis is 2a. Hence, the length of semi-major axis is a.

## Minor Axis

A minor axis is a line passing through the center and perpendicular to major axis of the hyperbola. The length of the minor axis is 2b. Hence, the length of semi-minor axis is b.

## Vertex

A vertex is the point at which major axis intersects the hyperbola. Again, vertices (plural of vertex) of a hyperbola are the two points of intersecton between the line segment joining to two foci and the hyperbola. Its distance from the center is a that is equal to the length of semi-major axis.

## Asymptote

An asymptote of a hyperbola is a straight line passes through the center and never touches the hyperbola.

## Focal Length

The distance of focal point from the center of a hyperbola is called the focal length or focal distance. The focal length is usually denoted by c

Different parts of hyperbola is shown in figure.

## Eccentricity

The ratio of focal length and the length of the semi-major axis is called the eccentricity of a hyperbola. Eccentricity is the amount of deviation for conic section from getting circular. It is denoted by a. If c is the focal length and a is the length of the semi-major axis, then eccentricity, e = ca. The eccentricity of a hyperbola is always greater than 1.

## Directrix

A directrix of a hyperbola is a straight line at a distance a^{2}c from the center and perpendicular to major axis where a and c are the length of the semi-major axix and focal distance of the parabola. Thus, a hyperbola has just two directrices.

## Latus Rectum

The latus rectum of a hyperbola is a line segment passes through the focus whose endpoins are on the hyperbola and perpendicular to the major-axis of the parabola. It is a special chord of the hyperbola passing through the focal point. It is also a unique chord passing through the focus and bisected itself by the axis of symmetry of the hyperbola.

## Equation of Hyperbola

x^{2}a^{2} - y^{2}b^{2} = 1