Parabola | Wikipedia audio article
Channel: wikipedia tts
Date: 03/10/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Parabola


00:02:36 1 History
00:04:33 2 Definition as a locus of points
00:06:18 3 In a cartesian coordinate system
00:06:29 3.1 Axis of symmetry parallel to the y-axis
00:13:16 3.2 General case
00:17:07 4 As a graph of a function
00:25:15 5 Similarity to the unit parabola
00:29:07 6 As a special conic section
00:30:14 7 In polar coordinates
00:34:14 8 Conic section and quadratic form
00:34:25 8.1 Diagram, description, and definitions
00:36:14 8.2 Derivation of quadratic equation
00:39:36 8.3 Focal length
00:39:55 8.4 Position of the focus
00:40:09 8.5 Alternative proof with Dandelin spheres
00:40:19 9 Proof of the reflective property
00:41:21 9.1 Construction and definitions
00:41:32 9.2 Deductions
00:41:48 9.3 Other consequences
00:42:43 9.3.1 Tangent bisection property
00:42:56 9.3.2 Intersection of a tangent and perpendicular from focus
00:43:06 9.3.3 Reflection of light striking the convex side
00:43:40 9.4 Alternative proofs
00:45:54 10 Pin and string construction
00:46:09 11 Properties related to Pascal's theorem
00:46:38 11.1 4-points-property
00:47:18 11.2 3-points-1-tangent-property
00:47:48 11.3 2-points-2-tangents-property
00:48:33 11.4 Axis-direction
00:49:45 12 Steiner generation
00:51:22 12.1 Parabola
00:52:54 12.2 Dual parabola
00:58:07 13 Inscribed angles and the 3-point-form
01:03:12 14 Pole-polar relation
01:07:36 15 Tangent properties
01:12:17 15.1 Two tangent properties related to the latus rectum
01:12:27 15.2 Orthoptic property
01:16:41 15.3 Lambert's theorem
01:19:41 16 Facts related to chords
01:32:06 16.1 Focal length calculated from parameters of a chord
01:38:59 16.2 Area enclosed between a parabola and a chord
01:39:09 16.3 Corollary concerning midpoints and endpoints of chords
01:39:48 17 Arc length
01:40:10 18 A geometrical construction to find a sector area
01:40:46 19 Focal length and radius of curvature at the vertex
01:40:56 20 As the affine image of the unit parabola
01:42:05 21 As quadratic Bézier curve
01:42:14 22 Numerical integration
01:42:37 23 As plane section of quadric
01:42:42 24 As trisectrix
01:42:54 25 Generalizations
01:45:57 26 In the physical world
01:47:13 26.1 Gallery
01:53:24 27 See also
01:56:00 28 Footnotes
02:07:00 29 Citations
02:10:48 30 Further reading
02:12:11 31 External links
02:12:42 As trisectrix
02:17:38 Generalizations
02:20:14 xp for other values of p are traditionally referred to as the higher parabolas, and were originally treated implicitly, in the form xp
02:20:45 xp/q for a positive fractional power of x. Negative fractional powers correspond to the implicit equation xpyq
02:21:28 In the physical world
02:26:48 Gallery
02:27:00 See also
02:27:42 Footnotes
02:27:51 Citations
02:28:00 Further reading
02:28:18 External links



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SUMMARY
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In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U shaped. It fits several other superficially different mathematical descriptions, which can all be proved to define exactly the same curves.

One description of a parabola involves a point (the focus) and a line (the directrix). The focus does not lie on the directrix. The parabola is the 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 which is parallel to another plane that is tangential to the conical surface.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 parabola that intersects the axis of symmetry is called the "vertex", and is the point where the parabola is most sharply curved. The distance ...