A conic with eccentricity greater than 1. Thus it is the locus of all points P such that the distance from P to a fixed point F₁ (the focus) is equal to e (>1) times the distance from P to a fixed line l₁ (the directrix). It turns out that there is another point F₂ and another line l₂ such that the same set of points would be obtained with these as focus and directrix. The hyperbola is also the conic section that results when a plane cuts a (double) cone in such a way that a section in two separate parts is obtained.

The line through F₁ and F₂ is the transverse axis, and the points V₁ and V₂ where it cuts the hyperbola are the vertices. The length |V₁V₂| is the length of the transverse axis and is usually taken to be 2a. The midpoint of V₁V₂ is the centre of the hyperbola. The line through the centre perpendicular to the transverse axis is the conjugate axis. It is usual to introduce b (>0) defined by b² = a²(e²−1), so that e²=1 + b²/a². The two separate parts of the hyperbola are the two branches.

A hyperbola in normal form

If a coordinate system is taken with origin at the centre of the hyperbola, and x‐axis along the transverse axis, the foci have coordinates (ae, 0) and (−ae, 0), the directrices have equations x = a/e and x = −a/e, and the hyperbola’s equation has the normal form

The hyperbola can be parametrized by x = asecθ‎, y = btanθ‎ (0 ≤ θ‎<2π‎, θ‎ ≠ π‎/2, 3π‎/2) or alternatively by x = acosh t, y = bsinht (tϵ‎ℝ) (see hyperbolic function), but this gives only one branch of the hyperbola.

The hyperbola, in normal form, has two asymptotes, y = (b/a)x and y = (−b/a)x. The shape of the hyperbola is determined by the eccentricity. The particular value $e=\sqrt{2}$ is important for this gives b = a. Then the asymptotes are perpendicular, and the hyperbola is called a rectangular hyperbola.