The parabola graph. Drag the sliders to change the parameters.
Parabola (mathematics) is a curve that is the most frequently defined by equation

(other alternative forms exist). The parabola is widely met in various areas of mathematics, physics and engineering. A free falling body in the gravity field (like a thrown ball) follows parabola trajectory (with a < 0 - branches down). A charged particle in electric field also follows parabolic trajectory.

The equation above assumes usual (cartesian) coordiates. The sigh of the parameter a defines the direction of branches (negative value - branches down, zero value - line equation), and the parameters b and c together define the offset of parabola in the coordinate space.


A parabola has a single axis of symmetry, which passes through its focus. The point of intersection of this axis and the parabola is called the vertex. A parabola spun about this axis in three dimensions traces out a shape known as a paraboloid of revolution. The derivative of parabola (a x + b) is a line. Parabola is a vertical cross section of cone.

Polar coordinates

In polar coordinates, a parabola with the focus at the origin and the directrix parallel to the y-axis, is given by the equation

where l is the semilatus rectum: the distance from the focus to the parabola itself, measured along a line perpendicular to the axis. Note that this is twice the distance from the focus to the vertex of the parabola or the perpendicular distance from the focus to the latus rectum.

The latus rectum is the chord that passes through the focus and is perpendicular to the axis. It has a length of 2l.

The earliest known work on conic sections was by Menaechmus in the fourth century B.C.. He discovered a way to solve the problem of doubling the cube using parabolae. (The solution, however, does not meet the requirements imposed by compass and straightedge construction.) The name "parabola" is due to Apollonius, who discovered many properties of conic sections. The focus–directrix property of the parabola and other conics is due to Pappus.

Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity.

The idea of using a paraboloid in a reflecting telescope is due to James Gregory in 1663 and the first to be constructed was by Isaac Newton in 1668. The same principle is used in satellite dishes and radar receivers.


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