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| {{for|other uses|Parabola (disambiguation)}}
| | The name of the author is Figures but it's not the most masucline name out there. To collect cash is what his family members and him enjoy. Years in the past we moved to North Dakota. I am a meter reader.<br><br>Also visit my website :: healthy food delivery ([http://www.myprgenie.com/view-publication/weight-loss-tips-that-will-reduce-your-waistline extra resources]) |
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| [[File:Parts of Parabola.svg|thumb|right|upright=1.36|Part of a parabola (blue), with various features (other colours). The complete parabola has no endpoints. In this orientation, it extends infinitely to the left, right, and upward.]]
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| A '''parabola''' ({{IPAc-en|p|ə|ˈ|r|æ|b|ə|l|ə}}; plural ''parabolas'' or ''parabolae'', adjective ''parabolic'', from {{lang-el|παραβολή}}) is a two-dimensional, [[Reflection symmetry|mirror-symmetrical]] [[curve]], which is approximately U-shaped when oriented as shown in the diagram, but which can be in any orientation in its [[Plane (geometry)|plane]]. It fits any of several superficially different [[Mathematics|mathematical]] descriptions which can all be proved to define curves of exactly the same [[shape]].
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| One description of a parabola involves a [[Point (geometry)|point]] (the [[Focus (geometry)|focus]]) and a [[Line (geometry)|line]] (the [[Directrix (conic section)|directrix]]). The focus does not lie on the directrix. The parabola is the [[locus (mathematics)|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 (geometry)|plane]] which is [[Parallel (geometry)|parallel]] to another plane which is [[Tangent|tangential]] to the conical surface.{{efn|The tangential plane just touches the conical surface along a line which passes through the apex of the cone}} A third description is [[algebra]]ic. A parabola is a [[Graph of a function|graph]] of a [[quadratic function]], such as <math>y=x^2.</math>
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| 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 axis of symmetry that intersects the parabola is called the "[[vertex (curve)|vertex]]", and it is the point where the [[curvature]] is greatest. 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 [[Similarity (geometry)|similar]].
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| Parabolas have the property that, if they are made of material that [[Reflection (physics)|reflects]] [[light]], then light which enters a parabola travelling parallel to its axis of symmetry 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 ("[[collimated]]") into a parallel 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.
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| 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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| Strictly, the adjective ''parabolic'' should be applied only to things that are shaped as a parabola, which is a two-dimensional shape. However, as shown in the last paragraph, the same adjective is commonly used for three-dimensional objects, such as [[parabolic reflector]]s, which are really [[paraboloid]]s. Sometimes, the noun ''parabola'' is also used to refer to these objects. Though not perfectly correct, this usage is generally understood.
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| {{clear}}
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| {{tocleft}}
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| [[File:Quadratic function.png|thumb|right|200px|Parabolic graph of quadratic function y=6x<sup>2</sup>+4x-8]]
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| [[File:Conic Sections.svg|right|thumb|200px|The parabola is a member of the family of [[conic section]]s]]
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| [[Image:Parabola with focus and directrix.svg|right|thumb|280px|Parabolic curve showing [[Directrix (conic section)|directrix]] (L) and focus (F). The distance from any point on the parabola to the focus (P<sub>n</sub>F) equals the [[perpendicular distance]] from the same point on the parabola to the directrix (P<sub>n</sub>Q<sub>n</sub>).]]
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| {{clr}}
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| [[Image:Parabola with focus and arbitrary line.svg|right|thumb|280px|Parabolic curve showing chord (L), focus (F), and vertex (V). L is an arbitrary [[Chord (geometry)|chord]] of the parabola perpendicular to its axis of symmetry, which passes through V and F. (The ends of the chord are not shown here.) The lengths of all paths Q<sub>n</sub> - P<sub>n</sub> - F are the same, equalling the distance between the chord L and the directrix. (See previous diagram above.) This is similar to saying that a parabola is an [[ellipse]], but with one focal point at infinity. It also directly implies, by the wave nature of light, that parallel light arriving along the lines Q<sub>n</sub> - P<sub>n</sub> will be reflected to converge at F. A linear wavefront along L is concentrated, after reflection, to the one point where all parts of it have travelled equal distances and are in phase, namely F. No consideration of angles is required.]]
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| [[Image:Parabel som keglesnit.jpg|left|thumb|280px|A parabola obtained as the intersection of a cone with a (red) plane parallel to a (checkered) plane which is tangential to the cone's surface.]]
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| {{clear}}
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| ==History==
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| [[File:Leonardo parabolic compass.JPG|left|thumb|180px|Parabolic compass designed by [[Leonardo da Vinci]]]]
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| The earliest known work on conic sections was by [[Menaechmus]] in the fourth century BC. He discovered a way to solve the problem of [[doubling the cube]] using parabolas. (The solution, however, does not meet the requirements imposed by [[Compass and straightedge constructions|compass and straightedge construction]].) The area enclosed by a parabola and a line segment, the so-called "parabola segment", was computed by [[Archimedes]] via the [[method of exhaustion]] in the third century BC, in his ''[[The Quadrature of the Parabola]].'' The name "parabola" is due to [[Apollonius of Perga|Apollonius]], who discovered many properties of conic sections. It means "application", referring to "application of areas" concept, that has a connection with this curve, as Apollonius had proved.<ref>[http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=196&bodyId=202 Apollonius' Derivation of the Parabola] at [http://mathdl.maa.org/convergence/1/ Convergence]</ref> The focus–directrix property of the parabola and other conics is due to [[Pappus of Alexandria|Pappus]].
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| [[Galileo Galilei|Galileo]] showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity.
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| The idea that a [[parabolic reflector]] could produce an image was already well known before the invention of the [[reflecting telescope]].<ref>{{cite book
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| |title=Reflecting Telescope Optics: Basic design theory and its historical development
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| |edition=2
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| |first1=Ray N.
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| |last1=Wilson
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| |publisher=Springer
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| |year=2004
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| |isbn=3-540-40106-7
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| |page=3
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| |url=http://books.google.com/books?id=PuN7l2A2uzQC}}, [http://books.google.com/books?id=PuN7l2A2uzQC&pg=PA3 Extract of page 3]
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| </ref> Designs were proposed in the early to mid seventeenth century by many [[mathematician]]s including [[René Descartes]], [[Marin Mersenne]],<ref>''Stargazer'', [http://books.google.com/books?id=2LZZginzib4C&pg=PA115&dq=mersenne+zucchi+parallel#PPA115,M1 p. 115].</ref> and [[James Gregory (mathematician)|James Gregory]].<ref>''Stargazer'', [http://books.google.com/books?id=2LZZginzib4C&pg=PA132&dq=Gregory++telescope+French+convex pp. 123 and 132]</ref> When [[Isaac Newton]] built the [[Newton's reflector|first reflecting telescope]] in 1668 he skipped using a parabolic mirror because of the difficulty of fabrication, opting for a [[spherical mirror]]. Parabolic mirrors are used in most modern reflecting telescopes and in satellite dishes and radar receivers.<ref>{{cite web
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| |url = http://farside.ph.utexas.edu/teaching/316/lectures/node136.html
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| |title = Spherical Mirrors
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| |first = Richard
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| |last = Fitzpatrick
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| |date = July 14, 2007
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| |work = Electromagnetism and Optics, lectures
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| |publisher = [[University of Texas at Austin]]
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| |at = Paraxial Optics
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| |accessdate = October 5, 2011
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| |separator =,
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| }}</ref>
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| {{clear}}
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| ==Equation in Cartesian coordinates==
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| Let the [[Directrix (conic section)|directrix]] be the line ''x'' = −''p'' and let the focus be the point (''p'', 0). If (''x'', ''y'') is a point on the parabola then, by Pappus' definition of a parabola, it is the same distance from the directrix as the focus; in other words:
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| :<math>x+p=\sqrt{(x-p)^2+y^2}. </math>
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| Squaring both sides and simplifying produces
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| :<math>y^2 = 4px\ </math>
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| as the equation of the parabola. By interchanging the roles of ''x'' and ''y'' one obtains the corresponding equation of a parabola with a vertical axis as
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| :<math>x^2 = 4py.\ </math>
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| The equation can be generalized to allow the vertex to be at a point other than the origin by defining the vertex as the point (''h'', ''k''). The equation of a parabola with a vertical axis then becomes
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| :<math>(x-h)^{2}=4p(y-k).\,</math>
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| The last equation can be rewritten
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| :<math>y=ax^2+bx+c\,</math>
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| so the graph of any function which is a polynomial of degree 2 in ''x'' is a parabola with a vertical axis.
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| More generally, a parabola is a curve in the [[Cartesian plane]] defined by an [[Irreducible polynomial|irreducible]] equation — one that does not factor as a product of two not necessarily distinct linear equations — of the general conic form
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| :<math> A x^{2} + B xy + C y^{2} + D x + E y + F = 0 \,</math>
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| with the parabola restriction that
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| :<math>B^{2} = 4 AC,\,</math>
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| where all of the coefficients are real and where ''A'' and ''C'' are not both zero. The equation is irreducible if and only if the determinant of the 3×3 matrix
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| :<math>\begin{bmatrix}
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| A & B/2 & D/2 \\
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| B/2 & C & E/2 \\
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| D/2 & E/2 & F
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| \end{bmatrix}.</math>
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| is non-zero: that is, if (''AC'' − ''B''<sup>2</sup>/4)''F'' + ''BED''/4 − ''CD''<sup>2</sup>/4 − ''AE''<sup>2</sup>/4 ≠ 0. The reducible case, also called the degenerate case, gives a pair of parallel lines, possibly real, possibly imaginary, and possibly coinciding with each other.<ref>Lawrence, J. Dennis, ''A Catalog of Special Plane Curves'', Dover Publ., 1972.</ref>
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| {{clear}}
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| ==Conic section and quadratic form==
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| {{Plain image with caption|image=Parabolic conic section.svg|caption=Cone with cross-sections}}
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| The diagram represents a [[cone]] with its axis vertical.{{efn|In the diagram, the axis is not exactly vertical. This is the result of a technical problem that occurs when a 3-dimensional model is converted into a 2-dimensional image. Readers should imagine the cone rotated slightly clockwise, so the axis, {{overline|AV}}, is vertical.}} The point A is its [[apex (geometry)|apex]]. A horizontal [[Cross section (geometry)|cross-section]] of the cone passes through the points B, E, C, and D. This cross-section is circular, but appears [[elliptical]] when viewed obliquely, as is shown in the diagram. An inclined cross-section of the cone, shown in pink, is inclined from the vertical by the same angle, '''''θ''''', as the side of the cone. According to the definition of a parabola as a conic section, the boundary of this pink cross-section, EPD, is a parabola. The cone also has another horizontal cross-section, which passes through the vertex, P, of the parabola, and is also circular, with a radius which we will call '''''r'''''. Its centre is V, and {{overline|PK}} is a diameter. The chord {{overline|BC}} is a diameter of the lower circle, and passes through the point M, which is the midpoint of the chord {{overline|ED}}. Let us call the lengths of the line segments {{overline|EM}} and {{overline|DM}} '''''x''''', and the length of {{overline|PM}} '''''y'''''.
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| Thus:
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| :<math>BM=2y\sin{\theta}.</math> (The triangle BPM is [[isosceles]].)
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| :<math>CM=2r.</math> (PMCK is a [[parallelogram]].)
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| Using the [[Power of a point#Theorems|intersecting chords theorem]] on the chords BC and DE, we get:
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| :<math>EM \cdot DM=BM \cdot CM</math>
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| Substituting:
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| :<math>x^2=4ry\sin{\theta}</math>
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| Rearranging:
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| :<math>y=\frac{x^2}{4r\sin{\theta}}</math>
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| For any given cone and parabola, ''r'' and ''θ'' are constants, but ''x'' and ''y'' are variables which depend on the arbitrary height at which the horizontal cross-section BECD is made. This last equation is a simple quadratic one which describes how ''x'' and ''y'' are related to each other, and therefore defines the shape of the parabolic curve. This shows that the definition of a parabola as a conic section implies its definition as the graph of a quadratic function. Both definitions produce curves of exactly the same shape.
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| ===Focal length===
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| It is proved [[#Axis of symmetry.2C focal length.2C and directrix|below]] that if a parabola has an equation of the form ''y'' = ''ax''{{super|2}}, where ''a'' is a constant, then <math>a=\frac{1}{4f},</math> where ''f'' is its focal length. Comparing this with the last equation above shows that the focal length of the above parabola is '''''r'' sin ''θ'''''.
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| ===Position of the focus===
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| If a line is perpendicular to the plane of the parabola and passes through the centre, V, of the horizontal cross-section of the cone passing through P, then the point where this line intersects the plane of the parabola is the focus of the parabola, which is marked F on the diagram. Angle VPF is [[Complementary angles|complementary]] to ''θ'', and angle PVF is complementary to angle VPF, therefore angle PVF is ''θ''. Since the length of {{overline|PV}} is ''r'', this construction correctly places the focus on the axis of symmetry of the parabola, at a distance ''r'' sin ''θ'' from its vertex.
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| ==Other geometric definitions==
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| A parabola may also be characterized as a conic section with an [[eccentricity (mathematics)|eccentricity]] of 1. As a consequence of this, all parabolas are [[Similarity (geometry)|similar]], meaning that while they can be different sizes, they are all the same shape. A parabola can also be obtained as the [[Limit of a function|limit]] of a sequence of [[ellipse]]s 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 [[Extended real number line|infinity]]. The parabola is an [[inversive geometry|inverse transform]] of a [[cardioid]].
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| 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.
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| The parabola is found in numerous situations in the physical world (see below).
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| ==Equations==
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| ===Cartesian===
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| In the following equations <math>h</math> and <math>k</math> are the coordinates of the vertex, <math>(h,k)</math>, of the parabola and <math>p</math> is the distance from the vertex to the focus and the vertex to the directrix.
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| ====Vertical axis of symmetry====
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| :<math>(x - h)^2 = 4p(y - k) \,</math>
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| :<math>y =\frac{(x-h)^2}{4p}+k\,</math>
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| :<math>y = ax^2 + bx + c \,</math>
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| where
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| :<math>a = \frac{1}{4p}; \ \ b = \frac{-h}{2p}; \ \ c = \frac{h^2}{4p} + k; \ \ </math>
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| :<math>h = \frac{-b}{2a}; \ \ k = \frac{4ac - b^2}{4a}</math>.
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| Parametric form:
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| :<math>x(t) = 2pt + h; \ \ y(t) = pt^2 + k \, </math>
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| ====Horizontal axis of symmetry====
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| :<math>(y - k)^2 = 4p(x - h) \,</math>
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| :<math>x =\frac{(y - k)^2}{4p} + h;\ \,</math>
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| :<math>x = ay^2 + by + c \,</math>
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| where
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| :<math>a = \frac{1}{4p}; \ \ b = \frac{-k}{2p}; \ \ c = \frac{k^2}{4p} + h; \ \ </math>
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| :<math>h = \frac{4ac - b^2}{4a}; \ \ k = \frac{-b}{2a}</math>.
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| Parametric form:
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| :<math>x(t) = pt^2 + h; \ \ y(t) = 2pt + k \, </math>
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| ====General parabola====
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| The general form for a parabola is
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| :<math>(\alpha x+\beta y)^2 + \gamma x + \delta y + \epsilon = 0 \,</math>
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| This result is derived from the general conic equation given below:
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| :<math>Ax^2 +Bxy + Cy^2 + Dx + Ey + F = 0 \, </math>
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| and the fact that, for a parabola,
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| :<math>B^2=4AC \,</math>.
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| The equation for a general parabola with a focus point ''F''(''u'', ''v''), and a directrix in the form
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| :<math>ax+by+c=0 \,</math>
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| is
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| :<math>\frac{\left(ax+by+c\right)^2}{{a}^{2}+{b}^{2}}=\left(x-u\right)^2+\left(y-v\right)^2 \,</math>
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| ===Latus rectum, semilatus rectum, and polar coordinates===
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| In [[coordinates (elementary mathematics)|polar coordinates]], a parabola with the focus at the origin and the directrix parallel to the ''y''-axis, is given by the equation
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| : <math>r (1 + \cos \theta) = l \,</math>
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| where ''l'' is the ''[[semilatus rectum]]'': the distance from the focus to the parabola itself, measured along a line perpendicular to the axis of symmetry. Note that this equals the [[perpendicular distance]] from the focus to the [[directrix (conic section)|directrix]], and is twice the focal length, which is the distance from the focus to the vertex of the parabola.
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| The latus rectum is the chord that passes through the focus and is perpendicular to the axis of symmetry. It has a length of 2''l''.
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| ===Gauss-mapped form===
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| A [[Gauss map|Gauss-mapped]] form:
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| <math>(\tan^2\phi,2\tan\phi)</math>
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| has normal
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| <math>(\cos\phi,\sin\phi)</math>.
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| ==Proof of the reflective property==
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| [[File:Parabel 2.svg|thumb|right|Reflective property of a parabola]]
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| The reflective property states that, if a parabola can reflect light, then light which enters it travelling parallel to the axis of symmetry is reflected to the focus. This is derived from the wave nature of light in the caption to a diagram near the top of this article. This derivation is valid, but may not be satisfying to readers who would prefer a mathematical approach. In the following proof, the fact that every point on the parabola is equidistant from the focus and from the directrix is taken as axiomatic.
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| Consider the parabola <math>y=x^2.</math> Since all parabolas are similar, this simple case represents all others. The right-hand side of the diagram shows part of this parabola.
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| '''Construction and definitions'''
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| The point ''E'' is an arbitrary point on the parabola, with coordinates <math>(x,x^2).</math> The focus is ''F'', the vertex is ''A'' (the origin), and the line ''FA'' (the y-axis) is the axis of symmetry. The line ''EC'' is parallel to the axis of symmetry, and intersects the x-axis at ''D''. The point ''C'' is located on the directrix (which is not shown, to minimize clutter). The point ''B'' is the midpoint of the line segment ''FC''.
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| '''Deductions'''
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| Measured along the axis of symmetry, the vertex, ''A'', is equidistant from the focus, ''F'', and from the directrix. Correspondingly, since ''C'' is on the directrix, the y-coordinates of ''F'' and ''C'' are equal in absolute value and opposite in sign. ''B'' is the midpoint of ''FC'', so its y-coordinate is zero, so it lies on the x-axis. Its x-coordinate is half that of ''E'', ''D'', and ''C'', i.e. <math>\frac{{x}}{{2}}.</math> The slope of the line ''BE'' is the quotient of the lengths of ''ED'' and ''BD'', which is <math>\frac{x^2}{\left(\frac{x}{2}\right)},</math> which comes to <math>2x.</math>
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| But <math>2x</math> is also the slope (first derivative) of the parabola at ''E''. Therefore the line ''BE'' is the tangent to the parabola at ''E''.
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| The distances ''EF'' and ''EC'' are equal because ''E'' is on the parabola, ''F'' is the focus and ''C'' is on the directrix. Therefore, since ''B'' is the midpoint of ''FC'', triangles ''FEB'' and ''CEB'' are congruent (three sides), which implies that the angles marked <math>\alpha</math> are congruent. (The angle above ''E'' is vertically opposite angle ''BEC''.) This means that a ray of light which enters the parabola and arrives at ''E'' travelling parallel to the axis of symmetry will be reflected by the line ''BE'' so it travels along the line ''EF'', as shown in red in the diagram (assuming that the lines can somehow reflect light). Since ''BE'' is the tangent to the parabola at ''E'', the same reflection will be done by an infinitesimal arc of the parabola at ''E''. Therefore, light that enters the parabola and arrives at ''E'' travelling parallel to the axis of symmetry of the parabola is reflected by the parabola toward its focus.
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| The point ''E'' has no special characteristics. This conclusion about reflected light applies to all points on the parabola, as is shown on the left side of the diagram. This is the reflective property.
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| ===Other consequences===
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| There are other theorems that can be deduced simply from the above argument.
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| ====Tangent bisection property====
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| The above proof, and the accompanying diagram, show that the tangent ''BE'' bisects the angle ''FEC''. In other words, the tangent to the parabola at any point bisects the angle between the lines joining the point to the focus, and perpendicularly to the directrix.
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| ====Intersection of a tangent and perpendicular from focus====
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| [[File:Parabola-antipodera.gif|thumb|right|200px|Perpendicular from focus to tangent]]
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| Since triangles ''FBE'' and ''CBE'' are congruent, ''FB'' is perpendicular to the tangent ''BE''. Since ''B'' is on the x-axis, which is the tangent to the parabola at its vertex, it follows that the point of intersection between any tangent to a parabola and the perpendicular from the focus to that tangent lies on the line that is tangential to the parabola at its vertex. See animated diagram..<ref name=ET>Tsukerman, Emmanuel, "On Polygons Admitting a Simson Line as Discrete Analogs of Parabolas", ''Forum Geometricorum'' 13 (2013), 197–208. [http://forumgeom.fau.edu/FG2013volume13/FG201321.pdf]</ref>
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| {{clear}}
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| ===Alternative proofs===
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| [[File:Parábola y tangente-prueba.svg|thumb|right|200px|Parabola and tangent]]
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| The above proofs of the reflective and tangent bisection properties use a line of calculus. For readers who are not comfortable with calculus, the following alternative is presented.
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| In this diagram, ''F'' is the focus of the parabola, and ''T'' and ''U'' lie on its directrix. ''P'' is an arbitrary point on the parabola. ''PT'' is perpendicular to the directrix, and the line ''MP'' bisects angle ''FPT''. ''Q'' is another point on the parabola, with ''QU'' perpendicular to the directrix. We know that ''FP''=''PT'' and ''FQ''=''QU''. Clearly, ''QT''>''QU'', so ''QT''>''FQ''. All points on the bisector ''MP'' are equidistant from ''F'' and ''T'', but ''Q'' is closer to ''F'' than to ''T''. This means that ''Q'' is to the "left" of ''MP'', i.e. on the same side of it as the focus. The same would be true if ''Q'' were located anywhere else on the parabola (except at the point ''P''), so the entire parabola, except the point ''P'', is on the focus side of ''MP''. Therefore ''MP'' is the tangent to the parabola at ''P''. Since it bisects the angle ''FPT'', this proves the tangent bisection property.
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| The logic of the last paragraph can be applied to modify the above proof of the reflective property. It effectively proves the line ''BE'' to be the tangent to the parabola at ''E'' if the angles <math>\alpha</math> are equal. The reflective property follows as shown previously.
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| ==Tangent properties==
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| ===Two tangent properties related to the latus rectum===
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| Let the line of symmetry intersect the parabola at point ''Q'', and denote the focus as point ''F'' and its distance from point ''Q'' as ''f''. Let the perpendicular to the line of symmetry, through the focus, intersect the parabola at a point ''T''. Then (1) the distance from ''F'' to ''T'' is 2''f'', and (2) a tangent to the parabola at point ''T'' intersects the line of symmetry at a 45° angle.<ref>Downs, J. W., ''Practical Conic Sections'', Dover Publ., 2003.</ref>{{rp|p.26}}
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| | |
| ===Orthoptic property===
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| [[File:Isoptic.png|thumb|right|Perpendicular tangents intersect on the directrix]]
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| {{main|Isoptic}}
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| If two tangents to a parabola are perpendicular to each other, then they intersect on the directrix. Conversely, two tangents which intersect on the directrix are perpendicular.
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| '''Proof'''
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| | |
| Without loss of generality, consider the parabola <math>y=x^2.</math> Suppose that two tangents contact this parabola at the points <math>(p,p^2)</math> and <math>(q,q^2).</math> Their slopes are <math>2p</math> and <math>2q,</math> respectively. Thus the equation of the first tangent is of the form <math>y=2px+C,</math> where <math>C</math> is a constant. In order to make the line pass through <math>(p,p^2),</math> the value of <math>C</math> must be <math>-p^2,</math> so the equation of this tangent is <math>y=2px-p^2.</math> Likewise, the equation of the other tangent is <math>y=2qx-q^2.</math> At the intersection point of the two tangents, <math>2px-p^2=2qx-q^2.</math> Thus <math>2x(p-q)=p^2-q^2.</math> Factoring the difference of squares, cancelling, and dividing by 2 gives <math>x=\frac{p+q}{2}.</math> Substituting this into one of the equations of the tangents gives an expression for the y-coordinate of the intersection point: <math>y=2p\left(\frac{p+q}{2}\right)-p^2.</math> Simplifying this gives <math>y=pq.</math>
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| We now use the fact that these tangents are perpendicular. The product of the slopes of perpendicular lines is −1, assuming that both of the slopes are finite. The slopes of our tangents are <math>2p</math> and <math>2q,</math>, so <math>(2p)(2q)=-1,</math> so <math>pq=-\frac{1}{4}.</math> Thus the y-coordinate of the intersection point of the tangents is given by <math>y=-\frac{1}{4}.</math> This is also the equation of the directrix of this parabola, so the two perpendicular tangents intersect on the directrix.
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| | |
| ===Lambert's theorem===
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| Let three tangents to a parabola form a triangle. Then '''Lambert's theorem''' states that the focus of the parabola lies on the circumcircle of the triangle.<ref name=ET/>{{rp|Corollary 20}}
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| ===Properties proved elsewhere in this article===
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| Click on link to find description and proof.
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| * [[#Other consequences|Tangent bisection property]]
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| * [[#Other consequences|Intersection of tangent and perpendicular from focus]]
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| * [[#Corollary concerning midpoints and endpoints of chords|Tangents at endpoints of chords]]
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| ==Dimensions of parabolas with axes of symmetry parallel to the y-axis==
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| These parabolas have equations of the form <math>y=ax^2+bx+c.</math> By interchanging <math>x</math> and <math>y,</math> the parabolas' axes of symmetry become parallel to the x-axis.
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| {{Plain image with caption|Parabola features.svg|Some features of a parabola|450px}}
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| ===Coordinates of the vertex===
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| The ''x''-coordinate at the vertex is <math>x=-\frac{b}{2a}</math>, which is found by [[derivative|differentiating]] the original equation <math>y=ax^2+bx+c</math>, setting the resulting <math>dy/dx=2ax+b</math> equal to zero (a [[critical point (mathematics)|critical point]]), and solving for <math>x</math>. Substitute this ''x''-coordinate into the original equation to yield:
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| :<math>y=a\left (-\frac{b}{2a}\right )^2 + b \left ( -\frac{b}{2a} \right ) + c.</math>
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| Simplifying:
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| :<math>=\frac{ab^2}{4a^2} -\frac{b^2}{2a} + c</math>
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| Put terms over a common denominator
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| :<math>=\frac{b^2}{4a} -\frac{2\cdot b^2}{2\cdot 2a} + c\cdot\frac{4a}{4a}</math>
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| :<math>=\frac{-b^2+4ac}{4a}</math>
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| :<math>=-\frac{b^2-4ac}{4a}=-\frac{D}{4a}</math>
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| where <math>D</math> is the [[discriminant]], <math>(b^2-4ac).</math>
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| Thus, the vertex is at point
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| :<math>\left (-\frac{b}{2a},-\frac{D}{4a}\right ).</math>
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| ===Coordinates of the focus===
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| Since the axis of symmetry of this parabola is parallel with the y-axis, the x-coordinates of the focus and the vertex are equal. The coordinates of the vertex are calculated in the preceding section. The x-coordinate of the focus is therefore also <math>-\frac{b}{2a}.</math>
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| To find the y-coordinate of the focus, consider the point, ''P'', located on the parabola where the slope is 1, so the tangent to the parabola at ''P'' is inclined at 45 degrees to the axis of symmetry. Using the reflective property of a parabola, we know that light which is initially travelling parallel to the axis of symmetry is reflected at ''P'' toward the focus. The 45-degree inclination causes the light to be turned 90 degrees by the reflection, so it travels from ''P'' to the focus along a line that is perpendicular to the axis of symmetry and to the y-axis. This means that the y-coordinate of ''P'' must equal that of the focus.
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| By differentiating the equation of the parabola and setting the slope to 1, we find the x-coordinate of ''P'':
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| :<math>y=ax^2+bx+c,</math>
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| :<math>\frac{dy}{dx}=2ax+b=1</math>
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| :<math>\therefore x=\frac{1-b}{2a}</math>
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| Substituting this value of <math>x</math> in the equation of the parabola, we find the y-coordinate of ''P'', and also of the focus:
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| :<math>y=a\left(\frac{1-b}{2a}\right)^2+b\left(\frac{1-b}{2a}\right)+c</math>
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| :<math>=a\left(\frac{1-2b+b^2}{4a^2}\right)+\left(\frac{b-b^2}{2a}\right)+c</math>
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| :<math>=\left(\frac{1-2b+b^2}{4a}\right)+\left(\frac{2b-2b^2}{4a}\right)+c</math>
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| :<math>=\frac{1-b^2}{4a}+c=\frac{1-(b^2-4ac)}{4a}=\frac{1-D}{4a}</math>
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| where <math>D</math> is the [[discriminant]], <math>(b^2-4ac),</math> as is used in the "Coordinates of the vertex" section.
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| The focus is therefore the point:
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| :<math>\left(-\frac{b}{2a},\frac{1-D}{4a}\right)</math>
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| ===Axis of symmetry, focal length, latus rectum, and directrix===
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| The above coordinates of the focus of a parabola of the form:
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| :<math>y=ax^2+bx+c</math>
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| can be compared with the coordinates of its vertex, which are derived in the section "Coordinates of the vertex", above, and are:
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| :<math>\left(\frac{-b}{2a},\frac{-D}{4a}\right)</math>
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| where <math>D=b^2-4ac.</math>
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| The axis of symmetry is the line which passes through both the focus and the vertex. In this case, it is vertical, with equation:
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| :<math>x=-\frac{b}{2a}</math>.
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| The focal length of the parabola is the difference between the y-coordinates of the focus and the vertex:
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| :<math>f=\left(\frac{1-D}{4a}\right)-\left(\frac{-D}{4a}\right)</math>
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| :<math>=\frac{1}{4a}</math>
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| It is sometimes useful to invert this equation and use it in the form: <math>a=\frac{1}{4f}.</math> See the section "[[#Conic section and quadratic form|Conic section and quadratic form"]], above.
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| The point where the slope of the parabola is 1 lies at one end of the latus rectum. The length of the semilatus rectum (half of the latus rectum) is the difference between the x-coordinates of this point, which is considered as ''P'' in the above derivation of the coordinates of the focus, and of the focus itself. Thus, the length of the semilatus rectum is:
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| :<math>\frac{1-b}{2a}+\frac{b}{2a}</math>
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| :<math>=\frac{1}{2a}</math>
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| :<math>=2f</math>, where <math>f</math> is the focal length.
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| The total length of the latus rectum is therefore four times the focal length.
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| Measured along the axis of symmetry, the vertex is the midpoint between the focus and the directrix. Therefore, the equation of the directrix is:
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| :<math>y=-\frac{D}{4a}-\frac{1}{4a}=-\frac{1+D}{4a}</math>
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| ==Area enclosed between a parabola and a chord==
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| {{plain image|File:Area between a parabola and a chord.svg|Parabola (magenta) and line (lower light blue) including a chord (blue). The area enclosed between them is in pink. The chord itself ends at the points where the line intersects the parabola.|200px|right|top|triangle|image override=<div style="width: 200px; height: 350px;"><div style="position: relative; left: -60px; top: -60px;">[[File:Area between a parabola and a chord.svg|290px|Parabola and line including chord.]]</div></div>}}</div>
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| <div style="position: relative; z-index: 2;">
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| The area enclosed between a parabola and a chord (see diagram) is two-thirds of the area of a parallelogram which surrounds it. One side of the parallelogram is the chord, and the opposite side is a tangent to the parabola.<ref>http://www.mathwarehouse.com/geometry/parabola/area-of-parabola.php</ref><ref>http://mysite.du.edu/~jcalvert/math/parabola.htm</ref> The slope of the other parallel sides is irrelevant to the area. Often, as here, they are drawn parallel with the parabola's axis of symmetry, but this is arbitrary.
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| A theorem equivalent to this one, but different in details, was derived by [[Archimedes]] in the 3rd Century BCE. He used the areas of triangles, rather than that of the parallelogram.{{efn|Archimedes proved that the area of the enclosed parabolic segment was 4/3 as large as that of a triangle that he inscribed within the enclosed segment. It can easily be shown that the parallelogram has twice the area of the triangle, so Archimedes' proof also proves the theorem with the parallelogram.}} See the article "[[The Quadrature of the Parabola]]".
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| If the chord has length '''''b''''', and is perpendicular to the parabola's axis of symmetry, and if the perpendicular distance from the parabola's vertex to the chord is '''''h''''', the parallelogram is a rectangle, with sides of '''''b''''' and '''''h'''''. The area, '''''A''''', of the parabolic segment enclosed by the parabola and the chord is therefore:
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| :<math>A=\frac{2}{3}bh</math>
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| This formula can be compared with the area of a triangle: <math>\frac{1}{2}bh</math>.
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| In general, the enclosed area can be calculated as follows. First, locate the point on the parabola where its slope equals that of the chord. This can be done with calculus, or by using a line that is parallel with the axis of symmetry of the parabola and passes through the midpoint of the chord. The required point is where this line intersects the parabola.{{efn|This method can be easily proved correct by calculus. It was also known and used by Archimedes, although he lived nearly 2000 years before calculus was invented.}} Then, using the formula given in the article "[[Perpendicular distance]]", calculate the perpendicular distance from this point to the chord. Multiply this by the length of the chord to get the area of the parallelogram, then by <math>\textstyle\frac{2}{3}</math> to get the required enclosed area.
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| ===Corollary concerning midpoints and endpoints of chords===
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| A corollary of the above discussion is that if a parabola has several parallel chords, their midpoints all lie on a line which is parallel to the axis of symmetry. If tangents to the parabola are drawn through the endpoints of any of these chords, the two tangents intersect on this same line parallel to the axis of symmetry.{{efn|A proof of this sentence can be inferred from the proof of the Orthoptic property, above. It is shown there that the tangents to the parabola y<nowiki>=</nowiki>x<sup>2</sup> at (p,p<sup>2</sup>) and (q,q<sup>2</sup>) intersect at a point whose x-coordinate is the mean of p and q. Thus if there is a chord between these two points, the intersection point of the tangents has the same x-coordinate as the midpoint of the chord.}}
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| </div>
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| {{clear}}
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| ==Length of an arc of a parabola==
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| If a point '''X''' is located on a parabola which has focal length <math>f,</math> and if <math>p</math> is the [[perpendicular distance]] from '''X''' to the axis of symmetry of the parabola, then the lengths of [[Arc (geometry)|arcs]] of the parabola which terminate at '''X''' can be calculated from <math>f</math> and <math>p</math> as follows, assuming they are all expressed in the same units.
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| :<math>h=\frac{p}{2}</math>
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| :<math>q=\sqrt{f^2+h^2}</math>
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| :<math>s=\frac{hq}{f}+f\ln\left(\frac{h+q}{f}\right)</math>
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| This quantity, <math>s</math>, is the length of the arc between '''X''' and the vertex of the parabola.{{efn|In this calculation, the [[square-root]], '''''q''''', must be positive. The quantity '''ln(''a'')''' is the [[natural logarithm]] of ''a'', i.e. its logarithm to base "e".}}
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| The length of the arc between '''X''' and the symmetrically opposite point on the other side of the parabola is <math>2s.</math>
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| The perpendicular distance, <math>p</math>, can be given a positive or negative sign to indicate on which side of the axis of symmetry '''X''' is situated. Reversing the sign of <math>p</math> reverses the signs of <math>h</math> and <math>s</math> without changing their absolute values. If these quantities are signed, '''the length of the arc between ''any'' two points on the parabola is always shown by the difference between their values of <math>s.</math>''' The calculation can be simplified by using the properties of logarithms:
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| :<math>s_1 - s_2 = \frac{h_1 q_1 - h_2 q_2}{f} +f \ln \left(\frac{h_1 + q_1}{h_2 + q_2}\right)</math>
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| | |
| This can be useful, for example, in calculating the size of the material needed to make a [[parabolic reflector]] or [[parabolic trough]].
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| | |
| This calculation can be used for a parabola in any orientation. It is not restricted to the situation where the axis of symmetry is parallel to the y-axis.
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| | |
| ==Mathematical generalizations==
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| In [[algebraic geometry]], the parabola is generalized by the [[rational normal curve]]s, which have coordinates <math>(x,x^2,x^3,\dots,x^n);</math> the standard parabola is the case <math>n=2,</math> and the case <math>n=3</math> is known as the [[twisted cubic]]. A further generalization is given by the [[Veronese variety]], when there is more than one input variable.
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| In the theory of [[quadratic form]]s, the parabola is the graph of the quadratic form <math>x^2</math> (or other scalings), while the [[elliptic paraboloid]] is the graph of the [[Definite bilinear form|positive-definite]] quadratic form <math>x^2+y^2</math> (or scalings) and the [[hyperbolic paraboloid]] is the graph of the [[indefinite quadratic form]] <math>x^2-y^2.</math> Generalizations to more variables yield further such objects.
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| The curves <math>y=x^p</math> for other values of ''p'' are traditionally referred to as the '''higher parabolas''', and were originally treated implicitly, in the form <math>x^p=ky^q</math> for ''p'' and ''q'' both positive integers, in which form they are seen to be algebraic curves. These correspond to the explicit formula <math>y=x^{p/q}</math> for a positive fractional power of ''x.'' Negative fractional powers correspond to the implicit equation <math>x^py^q=k,</math> and are traditionally referred to as '''higher hyperbolas.''' Analytically, ''x'' can also be raised to an irrational power (for positive values of ''x''); the analytic properties are analogous to when ''x'' is raised to rational powers, but the resulting curve is no longer algebraic, and cannot be analyzed via algebraic geometry.
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| ==Parabolas in the physical world==
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| In nature, approximations of parabolas and paraboloids are found in many diverse situations. The best-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]]).
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| | |
| The parabolic trajectory of projectiles was discovered experimentally by [[Galileo]] in the early 17th century, who performed experiments with balls rolling on inclined planes. He also later proved this [[mathematical]]ly in his book ''Dialogue Concerning Two New Sciences''.<ref>Dialogue Concerning Two New Sciences (1638) (The Motion of Projectiles: Theorem 1); see [http://oll.libertyfund.org]</ref>{{efn|However, this parabolic shape, as Newton recognized, is only an approximation of the actual elliptical shape of the trajectory, and is obtained by assuming that the gravitational force is constant (not pointing toward the center of the earth) in the area of interest. Often, this difference is negligible, and leads to a simpler formula for tracking motion.}} 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.
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| Another [[hypothetical]] situation in which parabolas might arise, according to the theories of physics described in the 17th and 18th Centuries by [[Sir Isaac Newton]], is in [[two-body orbit]]s; for example the path of a small planetoid or other object under the influence of the gravitation of the [[Sun]]. [[Parabolic orbit]]s do not occur in nature; simple orbits most commonly resemble [[hyperbola]]s or [[ellipse]]s. The parabolic orbit is the [[degeneracy (math)|degenerate]] intermediate case between those two types of ideal orbit. An object following a parabolic orbit would travel at the exact [[escape velocity]] of the object it orbits; objects in [[elliptical orbit|elliptical]] or [[hyperbolic orbit|hyperbolic]] orbits travel at less or greater than escape velocity, respectively. Long-period [[comet]]s travel close to the Sun's escape velocity while they are moving through the inner solar system, so their paths are close to being parabolic.
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| Approximations of parabolas are also found in the shape of the main cables on a simple [[suspension bridge]]. The curve of the chains of a suspension bridge is always an intermediate curve between a parabola and a [[catenary]], but in practice the curve is generally nearer to a parabola, and in calculations the second degree parabola is used.<ref name="Troyano">{{cite book
| |
| |title=Bridge engineering: a global perspective
| |
| |first1=Leonardo Fernández
| |
| |last1=Troyano
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| |publisher=Thomas Telford
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| |year=2003
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| |isbn=0-7277-3215-3
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| |page=536
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| |url=http://books.google.com/books?id=0u5G8E3uPUAC}}, [http://books.google.com/books?id=0u5G8E3uPUAC&pg=PA536 Chapter 8 page 536]
| |
| </ref><ref>{{cite book
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| |title=A memoir of suspension bridges
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| |first1=Charles Stewart
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| |last1=Drewry
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| |publisher=Oxford University
| |
| |year=1832
| |
| |isbn=
| |
| |page=159
| |
| |url=http://books.google.com/books?id=Nk-pQT7-EM4C}}, [http://books.google.com/books?id=Nk-pQT7-EM4C&pg=PA159 Extract of page 159]
| |
| </ref> Under the influence of a uniform load (such as a horizontal suspended deck), the otherwise catenary-shaped cable is deformed toward a parabola. Unlike an inelastic chain, a freely hanging spring of zero unstressed length takes the shape of a parabola. Suspension-bridge cables are, ideally, purely in tension, without having to carry other, e.g. bending, forces. Similarly, the structures of parabolic arches are purely in compression.
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| | |
| Paraboloids arise in several physical situations as well. The best-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 [[Focus (optics)|focal point]], or conversely, collimates light from a point source at the focus into a parallel beam. 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,<ref>{{cite journal
| |
| |last = Middleton
| |
| |first = W. E. Knowles
| |
| |date=December 1961
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| |title = Archimedes, Kircher, Buffon, and the Burning-Mirrors
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| |journal = Isis
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| |volume = 52
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| |issue = 4
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| |publisher = Published by: The University of Chicago Press on behalf of The History of Science Society
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| |location =
| |
| |pages = 533–543
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| |doi = 10.1086/349498
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| |jstor = 228646}}</ref> constructed parabolic mirrors to defend [[Syracuse, Italy|Syracuse]] against the [[Roman Empire|Roman]] fleet, by concentrating the sun's rays to set fire to the decks of the Roman ships. The principle was applied to [[telescope]]s in the 17th century. Today, paraboloid reflectors can be commonly observed throughout much of the world in [[microwave]] and satellite-dish receiving and transmitting antennas.
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| In [[parabolic microphone]]s, a parabolic reflector that reflects [[sound]], but not necessarily electromagnetic radiation, is used to focus sound onto a microphone, giving it highly directional performance.
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| 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]].
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| [[Aircraft]] used to create a [[zero gravity|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 [[Weightlessness|zero gravity]] for most purposes.
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| In the [[United States]], [[Geometric_design_of_roads#Profile|vertical curves]] in roads are usually parabolic by design.
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| ===Gallery===
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| Click on any image to enlarge it.
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| | |
| <gallery>
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| Image:Bouncing ball strobe edit.jpg|A bouncing ball captured with a stroboscopic flash at 25 images per second. Note that the ball becomes significantly non-spherical after each bounce, especially after the first. That, along with spin and [[air resistance]], causes the curve swept out to deviate slightly from the expected perfect parabola.
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| Image:ParabolicWaterTrajectory.jpg|Parabolic trajectories of water in a fountain.
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| File:Comet Kohoutek orbit p391.svg|The path (in red) of [[Comet Kohoutek]] as it passed through the inner solar system, showing its nearly parabolic shape. The blue orbit is the Earth's
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| Image:Ponte Hercilio Luz - Dezembro 1996 - by Sérgio Schmiegelow.jpg|[[Hercilio Luz Bridge]], [[Florianópolis]], [[Brazil]]. The supporting cables of [[suspension bridge]]s follow a curve which is intermediate between a parabola and a [[catenary]].<ref name="Troyano" />
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| File:Rainbow Bridge(2).jpg|The [[Rainbow Bridge (Niagara Falls)|Rainbow Bridge]] across the [[Niagara River]], connecting [[Canada]] (left) to the [[United States]] (right). The parabolic arch is in compression, and carries the weight of the road.
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| File:Celler de Sant Cugat lateral.JPG|Parabolic arches used in architecture
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| Image:Coriolis effect11.jpg|Parabolic shape formed by a liquid surface under rotation. Two liquids of different densities completely fill a narrow space between two sheets of transparent plastic. The gap between the sheets is closed at the bottom, sides and top. The whole assembly is rotating around a vertical axis passing through the centre. (See [[Rotating furnace]])
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| File:ALSOL.jpg|[[Solar cooker]] with [[parabolic reflector]]
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| File:Antenna 03.JPG|[[Parabolic antenna]]
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| File:ParabolicMicrophone.jpg|[[Parabolic microphone]] with optically transparent plastic reflector, used to overhear referee conversations at an American college football game.
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| File:Solar Array.jpg|Array of [[parabolic trough]]s to collect [[solar energy]]
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| File:Ed_d21m.jpg|[[Thomas Edison|Edison]]'s searchlight, mounted on a cart. The light had a parabolic reflector.
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| File:Physicist Stephen Hawking in Zero Gravity NASA.jpg|Physicist [[Stephen Hawking]] in an aircraft flying a parabolic trajectory to produce zero-gravity
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| </gallery>
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| ==Footnotes==
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| {{notelist}}
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| ==Citations==
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| {{reflist}}
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| ==Further reading==
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| * Lockwood, E. H. (1961): ''A Book of Curves'', Cambridge University Press
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| ==See also==
| |
| *[[Catenary]]
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| *[[Ellipse]]
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| *[[Hyperbola]]
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| *[[Dome#Parabolic dome|Parabolic dome]]
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| *[[Parabolic partial differential equation]]
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| *[[Parabolic reflector]]
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| *[[Paraboloid]]
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| *[[Quadratic equation]]
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| *[[Quadratic function]]
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| *[[Rotating furnace]], paraboloids produced by rotation
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| *[[Universal parabolic constant]]
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| ==External links==
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| {{Commons category|Parabolas}}
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| {{Wikisource1911Enc|Parabola}}
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| * {{springer|title=Parabola|id=p/p071150}}
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| * {{MathWorld|title=Parabola|urlname=Parabola}}
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| * [http://www.mathwarehouse.com/quadratic/parabola/interactive-parabola.php Interactive parabola-drag focus, see axis of symmetry, directrix, standard and vertex forms]
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| * [http://www.cut-the-knot.org/Curriculum/Geometry/ArchimedesTriangle.shtml Archimedes Triangle and Squaring of Parabola] at [[cut-the-knot]]
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| * [http://www.cut-the-knot.org/Curriculum/Geometry/ParabolaLambert.shtml Two Tangents to Parabola] at [[cut-the-knot]]
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| * [http://www.cut-the-knot.org/Curriculum/Geometry/ParabolaEnvelope.shtml Parabola As Envelope of Straight Lines] at [[cut-the-knot]]
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| * [http://www.cut-the-knot.org/Curriculum/Geometry/ParabolaMirror.shtml Parabolic Mirror] at [[cut-the-knot]]
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| * [http://www.cut-the-knot.org/Curriculum/Geometry/ThreeParabolaTangents.shtml Three Parabola Tangents] at [[cut-the-knot]]
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| *[http://math.fullerton.edu/mathews/n2003/TangentParabolaMod.html Module for the Tangent Parabola]
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| * [http://www.cut-the-knot.org/Curriculum/Geometry/ParabolaFocal.shtml Focal Properties of Parabola] at [[cut-the-knot]]
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| * [http://www.cut-the-knot.org/Curriculum/Geometry/ParabolaMesh.shtml Parabola As Envelope II] at [[cut-the-knot]]
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| * [http://dynamicmathematicslearning.com/similarparabola.html The similarity of parabola] at [http://dynamicmathematicslearning.com/JavaGSPLinks.htm Dynamic Geometry Sketches], interactive dynamic geometry sketch.
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| *[http://www.maverickexperiments.com/DrawConicSections/parabola.html A method of drawing a parabola with string and tacks]
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| [[Category:Conic sections]]
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| [[Category:Curves]]
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| [[Category:Parabolas]]
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