Dictionary Definition
parabola n : a plane curve formed by the
intersection of a right circular cone and a plane parallel to an
element of the curve
User Contributed Dictionary
English
Etymology
From (parabolē)Noun
 The conic section formed by the intersection of a cone with a plane parallel to a tangent plane to the cone; the locus of points equidistant from a fixed point (the focus) and line (the directrix).
Translations
The conic section formed by the intersection of
a cone with a plane parallel to a tangent plane to the cone
See also
Croatian
Noun
 parabola
Declension
Czech
Noun
Extensive Definition
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 parabola
can also be defined as the locus
of points
in a plane which are equidistant
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.
The parabola is an important concept in abstract
mathematics, but it is 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 yaxis.
 (x  h)^2 = 4p(y  k) \,
or, alternatively with axis parallel to the
xaxis
 (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
 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 \not= 0 \, or C \not= 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
does not 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 \,

 \mboxa = \frac; \ \ b = \frac; \ \ c = \frac + k; \ \
 h = \frac; \ \ k = \frac.
 x(t) = 2pt + h; \ \ y(t) = pt^2 + k \,
Horizontal axis of symmetry
 (y  k)^2 = 4p(x  h) \,
 x = a(y  k)^2 + h \,
 x = ay^2 + by + c \,

 \mboxa = \frac; \ \ b = \frac; \ \ c = \frac + h; \ \
 h = \frac; \ \ k = \frac.
 x(t) = pt^2 + h; \ \ y(t) = 2pt + k \, '''
General parabola
the general form for a parabola is: (Ax+By)^2 + Cx + Dy + E = 0 \,
Latus rectum, semilatus rectum, and polar coordinates
In
polar coordinates, a parabola with the focus at the origin and
the directrix on the positive xaxis, is given by the equation
 r (1 + \cos \theta) = l \,
where l is the semilatus
rectum: the distance from the focus to the parabola itself,
measured along a line perpendicular to the axis. Note that this is
twice the distance from the focus to the apex of the parabola or
the perpendicular distance from the focus to the latus
rectum.
The latus rectum is 4a.
Gaussmapped form
A Gaussmapped
form: (\tan^2\phi,2\tan\phi) has normal (\cos\phi,\sin\phi).
Derivation of the focus
Given a parabola parallel to the yaxis with
vertex (0,0) and with equation
 y = a x^2, \qquad \qquad \qquad (1)
Let F denote the focus, and let Q denote the
point at (x,f). Line FP has the same length as line QP.
 \ FP \ = \sqrt,
 \ QP \ = y + f.
 \ FP \ = \ QP \
 \sqrt = a x^2 + f \qquad
 x^2 + a^2 x^4 + f^2  2 a x^2 f = a^2 x^4 + f^2 + 2 a x^2 f \quad
 x^2  2 a x^2 f = 2 a x^2 f, \quad
 x^2 = 4 a x^2 f. \quad
 1 = 4 a f \quad
 f =
 x^2 = 4 p y \quad
All this was for a parabola centered at the
origin. For any generalized parabola, with its equation given in
the standard
form
 y=ax^2+bx+c,
the focus is located at the point
 \left (\frac,\frac+c+\frac \right)
which may also be written as
 \left (\frac,c\frac \right)
and the directrix is designated by the
equation
 y=\frac+c\frac
which may also be written as
 y=c\frac
Reflective property of the tangent
The tangent of the parabola described by equation
(1) has slope
 = 2 a x =
 F = (0,f), \quad
 Q = (x,f), \quad
 = = = (, 0).
 \ FG \ \cong \ GQ \,
 \ PF \ \cong \ PQ \,
 \Delta FGP \cong \Delta QGP
It follows that \angle FPG \cong \angle GPQ
.
Line QP can be extended beyond P to some point T,
and line GP can be extended beyond P to some point R. Then \angle
RPT and \angle GPQ are vertical,
so they are equal (congruent). But \angle GPQ is equal to \angle
FPG . Therefore \angle RPT is equal to \angle FPG .
The line RG is tangent to the parabola at P, so
any light beam bouncing off point P will behave as if line RG were
a mirror and it were bouncing off that mirror.
Let a light beam travel down the vertical line TP
and bounce off from P. The beam's angle of inclination from the
mirror is \angle RPT , so when it bounces off, its angle of
inclination must be equal to \angle RPT . But \angle FPG has been
shown to be equal to \angle RPT . Therefore the beam bounces off
along the line FP: directly towards the focus.
Conclusion: Any light beam moving vertically
downwards in the concavity of the parabola (parallel to the axis of
symmetry) will bounce off the parabola moving directly towards the
focus. (See parabolic
reflector.)
When b varies
Vertex of a parabola: Finding the ycoordinateWe know the xcoordinate at the vertex is
x=\frac, so substitute it into the equation y=ax^2+bx+c
 y=a\left (\frac\right )^2 + b \left ( \frac \right ) + c\qquad\textrm
 =\frac \frac + c
 =\frac \frac + c\cdot\frac
 =\frac
 =\frac=\frac
Thus, the vertex is at point…
 \left (\frac,\frac\right )
Parabolas in the physical world
In nature, approximations of parabolas and paraboloids are found in many diverse situations. The most wellknown 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 does not resemble a parabola.Another situation in which parabola may arise in
nature is in twobody 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 is 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 do not 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 wellknown 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.
References
External links
 Apollonius' Derivation of the Parabola at Convergence
 Interactive paraboladrag focus, see axis of symmetry, directrix, standard and vertex forms
 Archimedes Triangle and Squaring of Parabola at cuttheknot
 Two Tangents to Parabola at cuttheknot
 Parabola As Envelope of Straight Lines at cuttheknot
 Parabolic Mirror at cuttheknot
 Three Parabola Tangents at cuttheknot
 Module for the Tangent Parabola
 Focal Properties of Parabola at cuttheknot
 Parabola As Envelope II at cuttheknot
 Parabola Construction  An interactive sketch showing how to trace a parabola. (Requires Java.)
 Quadratic Bezier Construction  An interactive sketch showing how to trace the quadratic Bezier curve (a parabolic segment). (Requires Java.)
 More Interactive Parabola Construction (Javaenabled)
parabola in Afrikaans: Parabool
parabola in Arabic: قطع مكافئ
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parabola in Czech: Parabola (matematika)
parabola in Danish: Parabel
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parabola in Estonian: Parabool
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