hyperbolas

Hyperbola is a type of conic section formed by intersecting a double cone with a plane that cuts both nappes, and it can also be defined as the locus of points where the absolute difference of distances to two fixed points (the foci) is constant.

In the plane, a hyperbola consists of two disjoint curves called branches. The center is the midpoint between the foci; the axes of symmetry align with the coordinate axes for standard orientation. The branches approach straight lines called asymptotes.

Standard forms. In a Cartesian coordinate system with the center at the origin and the transverse axis along the x-axis, the equation is x^2/a^2 - y^2/b^2 = 1. The foci are at (±c,0) with c^2 = a^2 + b^2. The eccentricity e = c/a > 1. The asymptotes are y = ±(b/a)x. The vertical-transverse-axis form is y^2/a^2 - x^2/b^2 = 1, with foci at (0, ±c) and asymptotes x = ±(b/a)y.

Properties. The difference of distances from any point on the hyperbola to the two foci is constant and equals 2a. In the general second-degree form Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, a hyperbola satisfies B^2 - 4AC > 0.

Rotated hyperbolas. General conic equations can represent hyperbolas rotated with respect to the coordinate axes (B ≠ 0 in the general form).

Applications. Hyperbolas appear in astronomy and navigation, in the design of hyperbolic mirrors and antennas, and in orbital mechanics as loci of constant difference distance.