Concept

Hyperbola

A hyperbola is the set of all points in a plane such that the absolute value of the differences of the distances from the foci is constant.

Any hyperbola has two axes of symmetry: one that passes through both foci, called the transverse axis, and another that is perpendicular to the first and equidistant from both curves, called the conjugate axis.

The above graph is an example of a horizontal hyperbola. Notice that the transverse axis is a horizontal line. There are also vertical hyperbolas where the transverse axis is a vertical line. Below is a list of the properties of horizontal and vertical hyperbolas.

Property	Horizontal Hyperbola	Vertical Hyperbola
General Equation	$\frac{( x - h ) ^{2}}{a ^{2}} - \frac{( y - k ) ^{2}}{b ^{2}} = 1$	$\frac{( y - k ) ^{2}}{a ^{2}} - \frac{( x - h ) ^{2}}{b ^{2}} = 1$
Transverse Axis	Horizontal	Vertical
Vertices	$(h \pm a, k)$	$(h, k \pm a)$
Foci	$(h \pm c, k)$	$(h, k \pm c)$
Asymptotes	$y - k = \pm \frac{b}{a} (x - h)$	$y - k = \pm \frac{b}{a} (x - h)$
$c - value$	$c^{2} = a^{2} + b^{2}$	$c^{2} = a^{2} + b^{2}$

Although these curves are not considered to be parabolas, they do have foci and directrices as parabolas do.