parabol

Parabol is a plane curve defined as the set of all points equidistant from a fixed point, the focus, and a fixed line, the directrix. It is one of the conic sections and has eccentricity 1. The axis is the line through the focus perpendicular to the directrix; the vertex is the point closest to the directrix; the distance from the vertex to the focus is the focal length.

In Cartesian coordinates, a standard parabola opening to the right is y^2 = 4ax, where a > 0. Its focus is (a, 0) and its directrix is x = -a. Other orientations are obtained by translating or rotating the figure; for a parabola opening up, x^2 = 4ay has focus (0, a) and directrix y = -a. In general position, a parabola satisfies the condition B^2 = 4AC in the general quadratic form Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0.

Key properties include symmetry about the axis, a single branch, and the reflective property: rays parallel to the axis reflect toward the focus. This underpins the use of parabolic mirrors in telecommunications and astronomy, where parabolic reflectors concentrate energy at a single point.

Parabolas arise as the locus definition above, as a conic section from the intersection of a plane with a double cone parallel to a generator, and as graphs of quadratic functions. In three dimensions, the parabola can form a parabolic cylinder or a paraboloid when extended or rotated around an axis.