Parabola

A parabola is a plane curve that is the locus of all points equidistant from a fixed point, called the focus, and a fixed line, called the directrix. It is a type of conic section with eccentricity equal to 1.

In Cartesian coordinates, a non-rotated parabola with vertex at (h, k) and axis horizontal has equation (y − k)2 = 4p(x − h). Here p is the distance from the vertex to the focus; the focus is (h + p, k) and the directrix is the vertical line x = h − p. If the axis is vertical, the equation is (x − h)2 = 4p(y − k), with focus (h, k + p) and directrix y = k − p.

In standard forms, a vertical parabola can be written as y = ax2 + bx + c, and a horizontal one as x = ay2 + by + c. Rotated parabolas are described by the general second-degree equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, where B2 − 4AC = 0 characterizes a parabola.

Key features include the axis of symmetry, the vertex (the closest point to the focus on the parabola), and the focal length p, the distance from the vertex to the focus. The latus rectum is the line through the focus perpendicular to the axis, with length |4p|.

Parabolas arise as conic sections when a plane intersects a cone parallel to a generator. They have practical uses in optics and engineering, notably in satellite dishes, automobile headlights, and various reflecting devices, due to their property of focusing parallel rays to the focus.