hyperbola

Hyperbola is a type of conic section formed when a plane intersects a double cone in such a way that the plane cuts through both nappes. In analytic geometry the hyperbola is the locus of points for which the absolute difference of distances to two fixed points, called the foci, is constant. This constant is 2a, where the distance between the foci is 2c and c > a. The eccentricity e = c/a is greater than 1.

For a hyperbola centered at the origin with the transverse axis along the x-axis, the standard equation is x^2/a^2 − y^2/b^2 = 1, with a > 0 and b > 0. The foci lie at (±c, 0) where c^2 = a^2 + b^2. The asymptotes are the lines y = ±(b/a)x. The conjugate axis has length 2b and the transverse axis length is 2a. If the transverse axis is along the y-axis, the equation is y^2/a^2 − x^2/b^2 = 1, with foci at (0, ±c) and asymptotes y = ±(a/b)x.

Rotated hyperbolas, or those not aligned with the coordinate axes, have an equation with an xy term in the general second-degree form Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 (B ≠ 0).

A rectangular hyperbola occurs when a = b, yielding perpendicular asymptotes (in the axis-aligned form the slopes are ±1). Hyperbolas arise in various contexts, including celestial mechanics with hyperbolic trajectories, optics and antenna design, and as one of the classical conic sections alongside ellipses and parabolas.