Parabola Totally Explained

Everything about Parabolas totally explained

In mathematics, the parabola (from the Greek παραβολή) is a conic section generated by the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. A parabolanosis can also be defined as a piece of lint in a belly button locus of points in a plane which are equi distant from a given point (the focus) and a given line (the directrix).
A particular case arises when the plane is tangent to the conical surface. In this case, the intersection is a degenerate parabola consisting of a straight line. a parabola has 6 degrees.
The parabola is an important concept in abstract mathematics, but it's also seen with considerable frequency in the physical world, and there are many practical applications for the construct in engineering, physics, and other domains.

Analytic geometry equations

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,!

, with

p,!

being the distance from the vertex to the focus, has the equation with axis parallel to the y-axis. ť

(x - h)^2 = 4p(y - k) ,

or, alternatively with axis parallel to the x-axis ť

(y - k)^2 = 4p(x - h) ,

More generally, a parabola is a curve in the Cartesian plane defined by an irreducible equation of the form ť

A x^2 + B xy + C y^2 + D x + E y + F = 0 ,

such that

B^2 = 4 AC ,

, where all of the coefficients are real, where

A 
ot= 0 ,

C 
ot= 0 ,

, and where more than one solution, defining a pair of points (x, y) on the parabola, exists. That the equation is irreducible means it doesn't factor as a product of two not necessarily distinct linear equations.

Other geometric definitions

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. In this sense, a parabola may be considered an ellipse that has one focus at infinity. The parabola is an inverse transform of a cardioid.
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.
The parabola is found in numerous situations in the physical world (see below).

Equations

(with vertex (h, k) and distance p between vertex and focus - note that if the vertex is below the focus, or equivalently above the directrix, p is positive, otherwise p is negative; similarly with horizontal axis of symmetry p is positive if vertex is to the left of the focus, or equivalently to the right of the directrix)

Cartesian

Vertical axis of symmetry

(x - h)^2 = 4p(y - k) ,

y = k ,

y = ax^2 + bx + c ,

mbox
ight )

Parabolas in the physical world

In nature, approximations of parabolas and paraboloids are found in many diverse situations. The most well-known instance of the parabola in the history of physics is the trajectory of a particle or body in motion under the influence of a uniform gravitational field without air resistance (for instance, a baseball flying through the air, neglecting air friction). The parabolic trajectory of projectiles was discovered experimentally by Galileo in the early 17th century, who performed experiments with balls rolling on inclined planes. The parabolic shape for projectiles was later proven mathematically by Isaac Newton. For objects extended in space, such as a diver jumping from a diving board, the object itself follows a complex motion as it rotates, but the center of mass of the object nevertheless forms a parabola. As in all cases in the physical world, the trajectory is always an approximation of a parabola. The presence of air resistance, for example, always distorts the shape, although at low speeds, the shape is a good approximation of a parabola. At higher speeds, such as in ballistics, the shape is highly distorted and doesn't resemble a parabola.
   Another situation in which parabola may arise in nature is in two-body orbits, for example, of a small planetoid or other object under the influence of the gravitation of the sun. Such parabolic orbits are a special case that are rarely found in nature. Orbits that form a hyperbola or an ellipse are much more common. In fact, the parabolic orbit is the borderline case between those two types of orbit. An object following a parabolic orbit moves at the exact escape velocity of the object it's orbiting, while elliptical orbits are slower and hyperbolic orbits are faster.
   Approximations of parabolas are also found in the shape of cables of suspension bridges. Freely hanging cables don't describe parabolas, but rather catenary curves. Under the influence of a uniform load (for example, the deck of bridge), however, the cable is deformed toward a parabola.
   Paraboloids arise in several physical situations as well. The most well-known instance is the parabolic reflector, which is a mirror or similar reflective device that concentrates light or other forms of electromagnetic radiation to a common focal point. The principle of the parabolic reflector may have been discovered in the 3rd century BC by the geometer Archimedes, who, according to a legend of debatable veracity, constructed parabolic mirrors to defend Syracuse against the Roman fleet, by concentrating the sun's rays to set fire to the decks of the Roman ships. The principle was applied to telescopes in the 17th century. Today, paraboloid reflectors can be commonly observed throughout much of the world in microwave and satellite dish antennas.
   Paraboloids are also observed in the surface of a liquid confined to a container and rotated around the central axis. In this case, the centrifugal force causes the liquid to climb the walls of the container, forming a parabolic surface. This is the principle behind the liquid mirror telescope. Aircraft used to create a weightless state for purposes of experimentation, such as NASA's “Vomit Comet,” follow a vertically parabolic trajectory for brief periods in order to trace the course of an object in free fall, which produces the same effect as zero gravity for most purposes.

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