In mathematics, a

**parabola** is a plane curve, which is mirror-symmetrical, and is approximately U-shaped when oriented as shown in the diagram below (it remains a parabola if is differently oriented). It fits any of several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.

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In mathematics, a **parabola** is a plane curve, which is mirror-symmetrical, and is approximately U-shaped when oriented as shown in the diagram below (it remains a parabola if is differently oriented). It fits any of several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.

One description of a parabola involves a point (the focus) and a line (the directrix). The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane which is parallel to another plane which is tangential to the conical surface. A third description is algebraic. A parabola is a graph of a quadratic function, *y* = *x*^{2}, for example.

The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called the "axis of symmetry". The point on the parabola that intersects the axis of symmetry is called the "vertex", and is the point where the parabola is most sharply curved. The distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". The "latus rectum" is the chord of the parabola which is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some other arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola — that is, all parabolas are geometrically similar.

Parabolas have the property that, if they are made of material that reflects light, then light which travels parallel to the axis of symmetry of a parabola and strikes its concave side is reflected to its focus, regardless of where on the parabola the reflection occurs. Conversely, light that originates from a point source at the focus is reflected into a parallel ("collimated") beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other forms of energy. This reflective property is the basis of many practical uses of parabolas.

The parabola has many important applications, from a parabolic antenna or parabolic microphone to automobile headlight reflectors to the design of ballistic missiles. They are frequently used in physics, engineering, and many other areas.

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