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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 with an axis parallel to the y axis with vertex (h, k), focus (h, k + p), and directrix y = k - p has the equation

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. 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. See also parabolic reflector.

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 (Parametric):

Ellipse, Hyperbola, Paraboloid.