Newton skipped a parabolic mirror for his 1668 telescope because it was too hard to make
A parabolic mirror sends every ray parallel to its axis to one focus, which is why satellite dishes and most modern reflecting telescopes use that shape. Descartes, Mersenne and James Gregory all proposed such telescopes, but when Isaac Newton built the first reflector in 1668 he settled for a spherical mirror, which was easier to grind.
Several definitions all produce the same curve. Pick a point, the focus, and a line not passing through it, the directrix; the parabola is every point in the plane equally far from both. Alternatively, slice a right circular cone with a plane parallel to one that just touches the cone. Graph a quadratic function and you get one whose axis is vertical, and every such curve is the graph of a quadratic.
The axis of symmetry runs through the focus at right angles to the directrix and cuts the curve in half. It meets the curve at the vertex, the point of sharpest bend, and the distance from vertex to focus is the focal length. The chord through the focus parallel to the directrix is the latus rectum. A parabola may open in any direction, and because every parabola can be shifted and scaled to match any other, they are all geometrically similar.
The reflective property works both ways and for sound and other waves as well as light. Incoming rays parallel to the axis bounce to the focus from anywhere on the concave side, and a source placed at the focus leaves as a parallel beam. That explains parabolic antennas and microphones, car headlight reflectors and radar receivers, and the curve also figures in missile design and across physics and engineering.
Menaechmus, in the fourth century BC, wrote the earliest known study of conic sections and used parabolas to double the cube, though not by compass and straightedge. In the next century Archimedes found the area between a parabola and a chord by the method of exhaustion. Apollonius gave the curve its name, meaning application, after a link he proved with the classical idea of applying areas, and Pappus recorded the focus-directrix property. Galileo later showed that thrown objects trace parabolas because gravity accelerates them uniformly.
Source: Parabola
