Theory:

A curved mirror whose cross-section is shaped like the tip of a parabola is known as a parabolic mirror. It has a concave reflecting surface (curved inward), which directs all the incident beams of light to converge at its focal point.
 
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In this mirror, light rays produced by the source focused at the focal point will fall on the mirror's surface, and they will diverge in a direction, which is parallel to the principal axis of the parabolic mirror.
 
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Hence, it reflects the light rays to travel a long distance without getting diminished.

Parabolic mirrors are also known as parabolic reflectors, used to collect or project light energy, heat energy, sound energy and radio waves.
 
Applications:
 
They are used in,
  • Reflecting telescopes
  • Radio telescopes
  • Parabolic microphones
  • Solar cookers and
  • Solar water heaters, etc.,
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History:
 
The principle of parabolic reflectors has been known since Greco-Roman times.
  • Mathematician Diocles described and proved that parabolic reflectors focus a parallel beam to a point in his book "On Burning Mirrors".
  • Parabolic mirrors were also examined by the physicist Ibn Sahl in the \(10\)th century. In his book Optica Promota (\(1663\)), James Gregory expressed that a reflecting telescope with a parabolic mirror would correct spherical aberration and the chromatic aberration seen in refracting telescopes.
  • In 1888, the first parabolic mirrors were constructed by Heinrich Hertz, a German physicist, in the form of a reflector antenna.
Reference:
1. https://commons.wikimedia.org/wiki/File:Parabolic_Reflector_in_Nottingham_-_geograph.org.uk_-_1042997.jpg