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Parabola

A parabola is a conic section generated by the intersection of a cone and a plane parallel to some plane tangent to the cone. (If the plane is itself a tangent plane[?], one obtains a degenerate parabola consisting simply of a line.) A parabola may also be considered to be the set of points such that the distances of each point from a given point (the focus) and a given straight line (the directrix) are equal.

In Cartesian coordinates a parabola has the equation

[itex]y = ax^2[/itex]

with respect to some suitable coordinates.

A parabola may also be characterized as a conic section with an eccentricity of 1. As a consequence of this, all parabolas are similar. A parabola can also be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction.

A parabola has a single axis of reflective symmetry, which passes through its focus and is perpendicular to its directrix. A parabola spun about this axis in three dimensions traces out a shape known as a paraboloid of revolution. If a mirror is constructed in the shape of a paraboloid and a light source is placed at its focus, the light will be reflected as a beam of rays parallel to the axis, and the same process works in reverse. This device is called a parabolic reflector and finds applications in the construction of telescopes, spotlights, and LED housings. The same reflection principle is used in radio telescopes and parabolic microphones as well.

A particle in motion under the influence of a uniform gravitational field (for instance, a baseball flying through the air, neglecting air friction) follows a parabolic trajectory.

Equations (Cartesian):

[itex]y = ax^2[/itex]

Equations (Parametric):

[itex]x = 2at[/itex]
[itex]y = at^2[/itex]

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