What is Parabola: Definition and 362 Discussions

In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several 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 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 where the parabola intersects its axis of symmetry is called the "vertex" and is the point where the parabola is most sharply curved. 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 that 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 similar.
Parabolas have the property that, if they are made of material that reflects light, then light that travels parallel to the axis of symmetry of a parabola and strikes its concave side 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 into a parallel ("collimated") beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other waves. This reflective property is the basis of many practical uses of parabolas.
The parabola has many important applications, from a parabolic antenna or parabolic microphone to automobile headlight reflectors and the design of ballistic missiles. It is frequently used in physics, engineering, and many other areas.

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  1. C

    Can Geometry and Algebra Explain a Triangle's Romance with a Parabola?

    Homework Statement Here's the problem: refer to question no. 6 http://img1.uploadscreenshot.com/images/orig/5/12314183569-orig.jpg" Homework Equations I am not sure :S The Attempt at a Solution This, honestly, is a question which I have no clue how to go about... Thanks...
  2. A

    Finding a function for the parabola

    Note: I don't need any answer, all I want to know is whether this question is possible. Homework Statement "What is the function of the parabola which has the points (1, 1) (2, 2) and (3, 3)?" I just asked my teacher to get the question, It's not stated in my textbook. Homework...
  3. r-soy

    Quetion to find the equation of a parabola i want check

    Hi all Use the definition of a parabola and the distance formula to find the equation of a parabola with a ) directix x = -4 and focus (2,2 ) B ) directix x = 2 and focus (6,-4 ) my answer :
  4. r-soy

    Use the definition of a parabola and the distance

    Hi all Use the definition of a parabola and the distance formula to find the equation of a parabola with a ) directix x = -4 and focus (2,2 ) B ) directix x = 2 and focus (6,-4 ) How i solve like this queation please hle me the steps to solve that thanks
  5. r-soy

    Find the equaion of the parabola

    Find the equaion of the parabola having its vertex at the origin , its axis as indicated : A ) x-axis : (-4, - 20 ) b) y - axis : (30,-15 ) C) y - axis : (-root2,3) my answer A ) we have (-4, - 20 ) that mean the focus ( - 20 , 0 ) now we do comper 4ay...
  6. T

    Solve Absolute Parabola: Sketch |x^2 - 2x - 3| Graph

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  7. R

    Archimedes' rule that area of a parabola is 4/3 times an inscribed triangle

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  8. J

    Parabola for 18 aluminum mirror

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  9. L

    Find equation of parabola with focus and directrix

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  10. T

    Solve Parabola Equation: (10,0) ((13,27) (16,0)

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  11. L

    Mass of chain in shape of parabola

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  12. L

    Find Equation of Parabola Given Zero and Point

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  13. A

    Checking whether a curve is a parabola or not

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  14. M

    Parabola General to standard form

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  15. M

    Velocity vector along a parabola

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  16. G

    Apostol's Archimedes area proof for a parabola

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  17. C

    Find the equation of the tangent of the parabola y^2= 4px

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  18. D

    Solving for Acceleration on a Parabolic Path | Physics Tutorial

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  19. J

    Small Oscillations on a Parabola problem.

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  20. J

    What is the equation of a parabola?

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  21. Mentallic

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  22. M

    Parabola, tangebt line and Normal Line intersect

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  23. S

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  24. L

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  25. Y

    General form of a parabola

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  26. S

    Rotating a Parabola: 30o Anti-Clockwise About Origin

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  27. Z

    Parabola (Horizontal) - finding vertex, focus and equation of line

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  28. S

    ANOTHER parabola related question ._.

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  29. D

    Intersection of a parabola with another curve

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  30. S

    Is a Paper Strip Contracted on a Table a Parabola?

    When you take a paper strip, put it on the flat surface (table), and contract it so it stands up - is it parabola or something else?
  31. J

    Distance from vertex to focus in a parabola

    Homework Statement I am a bit rusty on parabolae. I am doing a question on projectiles and have found the coordinates of the vertex of a parabola as: (\frac{(v_0)^2\sin\alpha\cos\alpha}{g},\frac{(v_0)^2(\sin^2\alpha)}{2g}) The question now requires you to show that the distance...
  32. W

    Finding the function of a Parabola

    1. The graph of a quadratic function (a parabola) has x-intercepts -1 and 3 and a range consisting of all number less than or equal to 4. Determine an expression for the function. 2. none 3. I do not know how to use the intercepts and the range and manipulate it into a function...
  33. S

    Draw parabola using these materials

    You have a flat wooden surface, some rope, some nails and a pencil - how do you draw parabola using just that?
  34. D

    What Is the General Equation of a Parabola and Its Derivation?

    I read a Wikipedia article titled "Parabola" that listed an equation of the form: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 How is this derived? or where did it come from?
  35. S

    Circle inscribed in a parabola

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  36. M

    Determining the equation of an ellipse from its intersection with a parabola

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  37. R

    Exploring Limits at (0,0) on a Parabola

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  38. D

    Where is the vertex of the parabola f(x) = x^2 - 5x + 6?

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  39. C

    Find the equation of a parabola

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  40. U

    Transformation that takes all point on parabola onto unit circle

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  41. B

    Parametrizing and Line integrals (of a line, parabola, curve.)

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  42. K

    Shortest Distance on a Parabola: Point to Point Calculation

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  43. S

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  44. P

    What Part of a Ball Hitting a Parabola Hits the Focal Point?

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  45. J

    Standard form of the equation of parabola

    Homework Statement Write the equation in standard form for the parabola with an axis of symmetry y=0 and focus(-5,0). Homework Equations I think ax^2+bx+c. Also maybe x = -b/2a. But I don't know how to apply these. The Attempt at a Solution I know that focus and vertex lie on...
  46. turin

    Transforming parabola to straight line

    To the moderator: This isn't a HW question, but it probably sounds like one, so I appologize. Please move this to the HW forum if need be. I have an integration domain inside three intersecting curves. Two of the curves are straightlines and the third is a parabola. These three boundaries...
  47. P

    Tangents of a parabola perpendicular to a line

    A Tangent to the parabola y = 3x^2 - 7x + 5 is perpendicular to x + 5y - 10 = 0 Determine the equation of the tangent i really don't know where to start!
  48. E

    Projectile Motion parabola effect

    Homework Statement I'm still kind of confused with projectile motion. I know when you throw something, you throw it at an angle and will have a negative parabola effect. An exception to when you throw something up at a 90° angle where it will go up vertically and come down vertically, with no...
  49. I

    Vertical distance between two parabola

    Homework Statement Write a function for d(x), the vertical distance between the two curves, and find the minimum value of d(x). Homework Equations The equation for parabola one is y = x^2 + 6, for parabola two, y = -(x-2)^2 + 6 The Attempt at a Solution The answer in the back...
  50. C

    How to find the equation of a parabola with the following vertex.

    1. I need to find the equation of a parabola with a vertex of (-3,-2). It has to open down. 2. I understand of hopefully will understand it. 3. My answer was y=-3x Squared -3x-2 Which I found to be wrong.
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