What is Parabola: Definition and 361 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. brotherbobby

    A particle moving in a parabolic path in the ##x-y## plane

    Problem statement : I copy and paste the problem as it appears in the text down below. I have only changed the symbol of the given acceleration from ##a\rightarrow a_0##, owing to its constancy. Attempt : I must admit that I could proceed very little. Given...
  2. R

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    what i tried to do is to write y=v_0tsin alpha - 1/2gt^2 and x=v_0 cos alpha tand that t=x/v_0 cos alphai plug t in the formula for y and get that y= x tan alpha - gx^2/v_0^2 (tan^2 alpha -1)since jaan klada said there should be a quadratic equation (because its a parabola) i thought that...
  3. question_asker

    I Tracing parabolic motion with only current velocity and position?

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  4. H

    Kinematic Problem w/ Parabola: Solving w/ KE Theorem?

    This is not really a homework problem (it could be made to be though). I kind of made it up, inspired by a youtube math challenge problem involving parabolas, a water fountain where A = 1, R = 3, and H = 3. The solution given (h = 9/4) was based off simple math utilizing vertex form of a...
  5. rudransh verma

    B What is the focus and parameter of a parabola with vertex off the origin?

    The general eqn of parabola is ##(x-h)^2=-4a(y-k)##. This is the parabola whose vertex doesn't lie on origin and axis is parallel to y axis. It opens downwards. Vertex is (h,k). What will be the focus of this parabola and what is ##a## in general form? In the diagram a<0 which is...
  6. Istiak

    Find focal length of electron for a parabolic motion

    Here I was going to use ##\int \vec F \cdot d\vec l = \frac{1}{2}mu^2## What I got that is ##l=\frac{mu^2}{2eE}##. Here the question is what is ##l## (I took ##x## while doing the work but here I used ##l## instead of ##x##)? I was assuming that it's ##x## since I am calculating work in the...
  7. DaalChawal

    MHB Parabola Tangent: GP Relation for Fixed Point Chords

    Tangent is drawn at any point ( $x_1$ , $y_1$ ) other than vertex on the parabola $y^2$ = 4ax . If tangents are drawn from any point on this tangent to the circle $x^2$ + $y^2$ = $a^2$ such that all chords of contact pass through a fixed point ( $x_2$ , $y_2$ ) then (A) $x_1$ , a , $x_2$ are in...
  8. CallMeDirac

    How can I determine the parabola of a projectile?

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  9. rxh140630

    *solved*Particle moving along a parabola

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  10. karush

    MHB -gre.ge.04 intersection of parabola and line

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  11. E

    B How did Cavalieri get his formula for the area underneath a parabola?

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  12. Jalal_khan

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  13. A

    MHB Determine the equation of the parabola

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  14. F

    Parabola: Vertex =(?,0) and know 2 arbitrary points. How solve x?

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  15. R

    Intersection of a circle and a parabola

    We have a circle (x^2 + y^2=2) and a parabola (x^2=y). We put x^2 = y in the circle equation and we get y^+y-2=0. We get two values of y as y=1 and y=-2. Y=1 gives us two intersection point i.e (1,1) and (-1,1). But y=-2 neither it lie on the circle nor on the parabola. The discriminant of the...
  16. ElectronicTeaCup

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  17. minimoocha

    MHB Exploring How Archimedes Discovered Quadrature of the Parabola w/o Calculus

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  18. A

    B Parabola vs Hyperbola, why does a Hyperbola have two foci/curves?

    So I read a description saying something along the lines of, a Parabola does have a 2nd focus and directrix, but that they stretch off into infinity, whereas for the hyperbola the 2nd focus comes back round..? Anyway, I'm trying to picture it and understand in relation to the eccentricity, e...
  19. Kaushik

    Question involving calculations with a parabola

    Summary: Find the equation of the parabola when the focus and the equation of tangent at the vertex is known. Find the equation of the parabola when the focus is ##(0,0)## and the equation of tangent at the vertex is ##x - y + 1 = 0##. General equation : ## ax^2 + by^2 + 2hxy + 2gx + 2fy + c...
  20. S

    MHB Find vertex, focus, and directrix of parabola: y^2+12y+16x+68=0

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  21. DaveC426913

    B Parabola and Hyperbola Question

    Correct me if I'm wrong: A parabola extends without limit toward parallel lines. A hyperbola extends without limit toward diverging lines. They have very different equations. My question: is the former a specific instance of the latter? Does a parabola = a hyperbola that happens to have...
  22. Q

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  23. opus

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  24. Krushnaraj Pandya

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    Homework Statement Find the value of z for which (10x-5)^2 + (10y-7)^2 = z^2((5x+12y+7)^2 is a parabola Homework Equations eccentricity of parabola=1 The Attempt at a Solution I can solve this by expanding everything and writing h^2-ab=0 but this equation looks suspiciously similar to...
  25. J

    Acceleration of a particle on a parabola

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  26. YoungPhysicist

    B Really basic quadratic function problem

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  27. R

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  28. Poetria

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    Homework Statement ##r=\frac 1 {cos(\theta)+1}## y=-x A region bounded by this curve and parabola is to be found. 2. The attempt at a solution I have found the points of intersection but I am not sure what to do with the line (I need polar coordinates and it is not dependent on r :( )...
  29. D

    I Likelihood of the maximum of a parabola

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  30. Monoxdifly

    MHB Quadratic Equation Help: Find 3 Points to Determine Answer

    In a graph , straight line intersects the parabola at(-3,9) & (1, 1) Then the equation is A) x^2-2x+3=0 B) x^2+2x-3=0 C) x^2-3x+2=0 D) x^2-2x-3=0 I know that I can find the answer by substituting the known values to each options, but how to do it the proper way? We need at least three known...
  31. Monoxdifly

    MHB How to Determine the Reflection of a Parabola by a Given Line?

    Determine the reflection of a parabola y^2-2y-4x-11=0 by the line y = -x. I know how to do it graphically, but please tell me how to do it algebraically.
  32. Suyash Singh

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    Homework Statement A partice is projected at an angle 60 degrees with the speed 10 m/s. Then latus rectum is ? g= 10 m /s^2 Homework Equations i calculated the maximum height.Now what?? The Attempt at a Solution h= u u sin theta sin theta/2g
  33. Suyash Singh

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    Homework Statement The maximum range of a bullet fired from a toy pistol mounted on a car at rest is R0=40m . What will be the acute angle of inclination of the pistol for maximum range when the car is moving in the direction of firing with uniform v=velocity 20m/s, on a horizontal surface ...
  34. CCMarie

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  35. V

    B Prove that tangents to the focal cord of parabola....

    Prove that tangents to the focal cord of parabola are perpendicular using the reflection property of parabola ( A ray of light striking parallel to the focal plane goes through the focus, and a ray of light going through the focus goes parallel) I don't know whether this is solvable with just...
  36. T

    MHB Finding x of line bisecting parabola

    Hey, I have this question I've been trying to figure out in an integration textbook. The part of the question that I'm having trouble understanding is basically this. With the parabola below, find the x coordinate of A, if the line OA divides the shaded area into two equal parts. The area of...
  37. I

    Arc Length of Parabola & Square Root Function

    Homework Statement Consider the curves: y = x^2 from 1/2 to 2 and y = \sqrt{x} from 1/4 to 4. a. Explain why the lengths should be equal. b. Set up integrals (with respect to x) that give the arc lengths of the curve segments. Use a substitution to show that one integral can be...
  38. Ackbach

    MHB TiKZ Question on Plotting a Parabola

    I want to plot an hormetic curve. This is the code I have so far: \begin{tikzpicture}[scale = 0.75] %preamble \usepackage{pgfplots} \begin{axis} [xlabel=Exposure, ylabel=Benefit] \end{axis} \draw[black, line width = 0.50mm] plot[smooth,domain=0:6] (\x, {4-(\x-3)^2}); \end{tikzpicture}...
  39. H

    Find the equation of a tangent line to y = x^2?

    Homework Statement the line goes through (0, 3/2) and is orthogonal to a tangent line to the part of parabola y = x^2, x > 0 Homework EquationsThe Attempt at a Solution I have problems regarding finding the equation of tangent line to the part of parabola because the question not specifically...
  40. M

    MHB Finding the Equation of a Parabola

    Find the equation of a quadratic function whose graph contains the given points. (-2,1), (-6,1), (2,-7) THANK YOU
  41. H

    Calculating the equations of motion for particle in parabola

    I made the problem up myself, so there might very well not be a rational answer that I like! Homework Statement A point-particle is released at height h0 is released into a parabola. The position of the particle is given by (x, y) and the acceleration due to gravity is g. All forms of friction...
  42. J

    MHB Quadratics: How to determine parabola equation.

    Hi all. I need a bit of help determining the equation of some parabolas given points, intercepts and vertexes. Below are the exacts questions, any help will be much appreciated as I need this done soon! 1. A parabola has turning point (1,6) and passes through the point (-1,8). Find its...
  43. Bunny-chan

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    Homework Statement a) Show that for a given velocity V_0 a projectile can reach the same range R from two different angles \theta = 45 + \delta and \theta = 45 - \delta, as long as R doesn't go over the maximum range R_{max} = \frac{V_0^2}{g}. Calculate \delta in function of V_0 and R. b)...
  44. Tris Fray Potter

    B Can Parabolas Transform into Ellipses?

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  45. I

    MHB Where Should I Begin with Parabolas?

    I don't know where to start :s
  46. R

    I Is (u,v) = (x square - x, x+1) a Parametric Form of a Parabola?

    Hello. How can I verify that (u,v) = (x square - x, x+1) is a parametric form of a parabola? Thank you!
  47. Tris Fray Potter

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    HI! I'm not sure if this can go in precalculus or not because I'm from Australia, and our Maths subjects don't get that specific until university level. 1. Homework Statement For my assignment on quadratic functions, I have to find the equation (the the form of ax^2+bx+c) for a table of...
  48. INAM KHAN

    I Solving Equations of the Parabola

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  49. Theia

    MHB Finding Distinct and Singular Normals for a Parabola

    Let y = 2x^2 + 4x + \tfrac{7}{4} and line p a normal which go through the point (X, Y). Find the regions in xy-plane where there are either 3 distinct normals or only 1 normal.
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