Parabolas

Parabolas are planar curves defined as the set of all points equidistant from a fixed point, the focus, and a fixed line, the directrix. The axis of symmetry is the line through the focus perpendicular to the directrix, and it passes through the vertex, the parabola’s point closest to the focus.

Algebraically, parabolas can be described in several common forms. Vertical parabolas have equations of the form y = ax^2 + bx + c, with vertex at (h, k) in the equivalent vertex form y = a(x − h)^2 + k. Sideways parabolas have x = ay^2 + by + c or, in standard form, (y − k)^2 = 4p(x − h). The parameter p is the focal length: for vertical parabolas, the focus is (h, k + p) and the directrix is y = k − p; for horizontal parabolas, the focus is (h + p, k) and the directrix is x = h − p. When written as y = a(x − h)^2 + k, the focal length is p = 1/(4a).

Key properties include the axis of symmetry, x = h for vertical parabolas and y = k for horizontal ones; the vertex (h, k); and the focal length p, which determines the parabola’s width and direction. The length of the latus rectum, a line segment through the focus parallel to the directrix, has length |4p|.

Parabolas arise as conic sections (the locus of points at a fixed distance from a point and a line) and have widespread applications. They model the shape of satellite dishes and headlights, describe optical reflectors, and describe projectile trajectories in physics under uniform gravity. In mathematics, their special reflective property underpins many algorithms and designs.