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

    Jaan Kalda Kinematics question -- What regions can this cannon reach with its projectile?

    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?

    Is it possible to trace the trajectory of an object using only its velocity and position, both of which are given as components. My method of doing so involves using the time until max height is reached, and using that time value to calculate the max height itself (h,k), then plugging in the...
  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 General eqn of parabola

    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

    Parabola of a projectile?

    I have been puzzling over an equation that could be made to show the parabola of a projectile. So far I have determined that the lateral and vertical velocities are needed, the lateral velocity should determine the x² function but after that I am stuck. To specify I refused to look this up as...
  9. rxh140630

    *solved*Particle moving along a parabola

    Would the trivial solution be x=0,y=0? Non trivial: let y=x^2 \frac{dy}{dx}=2x, \frac{dx}{dy} = \frac12y^{-\frac12} x=\frac14 y^{-\frac12} here x=1 and y = 1/16 is a solution but my book says the answer is x=1/2 and y=1/4 this is one answer that you get with the equation I derived, but I...
  10. karush

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

    $\textbf{xy-plane}$ above shows one of the two points of intersection of the graphs of a linear function and and quadratic function. The shown point of intersection has coordinates $\textbf{(v,w)}$ If the vertex of the graph of the quadratic function is at $\textbf{(4,19)}$, what is the value of...
  11. E

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

    I know he had this ratio: But how did he get this: ?
  12. Jalal_khan

    Graphing to find the intersections of lines and a parabola in this limit

    Hello, I am currently in my college holidays and I have decided to do some maths to improve. My weakness is graphing and I am hoping to get some help or the solution on this question. Question: Let P(k,k^2) be a point on the parabola y=x^2 with k>0. Let O denote the origin. Let A(0, a)denote...
  13. A

    MHB Determine the equation of the parabola

    Determine the equation of the parabola with range y|y≧-6 and x-intercepts at -5 and 3.
  14. F

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

    Summary:: I have a upwards opening parabola where I know the Y vertex point = 0. I also know 2 arbitrary points separated by a step size on the parabola. How can I solve for x at the vertex point? X=horizontal plane Y=vertical plane I have a upwards opening parabola where I know the Y vertex...
  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

    Tension T in a parabolic wire at any point

    I am unsure how to go about this. I tried following the suggestion blindly and end up with with some cumbersome terms that are not the answer. From what I understand the derivative at each point would equal to T? Answer: I just can seem to get to this. I think I'm there but can't get it in...
  17. minimoocha

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

    How did Archimedes discover the Quadrature of the parabola without the use of calculus? If someone could please explain, I would be eternally grateful.
  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

    Find the vertex, focus, and directrix of the following parabola: y^2+12y+16x+68=0 The form we have been using is (y-k)^2=4p(x-h) Any explanation would help too, I'm really stuck on this one. Thank you!
  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

    Disc going along a parabola

    Homework Statement A disc of radius R rolls without slipping along the parabola y= ax2. Obtain the constrain equation Homework Equations Because there's no slipping, then: ##R d \theta = ds (1)## Where ##\theta ## is the angle between the line from the center of the disc to a fixed point...
  23. opus

    Motion throughout a parabola

    Homework Statement You throw a ball straight upward with initial speed ##v_0##. How long does it take to return to your hand? Homework Equations ##v=v_0+at## ##x-x_0=v_0t+\frac{1}{2}at^2## The Attempt at a Solution My question is a general one that relates to the described solution. According...
  24. Krushnaraj Pandya

    Identifying the equation of a parabola

    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

    Homework Statement A particle moves along a parabola on the x-y plane with equation ##y^{2}=2px## with constant speed ##1000m/s##.What is the magnitude of its acceleration? Homework Equations Parametric equations ##\vec{r}=(b^{2}t^{2}/(2p),bt)##. The Attempt at a Solution...
  26. YoungPhysicist

    B Really basic quadratic function problem

    I can't come up with this function for hours: I just started to learn quadratic functions so there must be an obvious solution for this which I clearly missed...
  27. R

    Condition of tangency of a line on a general form parabola

    <Moderator's note: Moved from a technical forum and thus no template. Effort in post #3.> What is the condition of tangency of a line y=mx+c on parabola with vertex(h,k) ,say for parabola (y-k)2=4a(x-h)? I could only find the condition of tangency on standard form of parabola, in the internet...
  28. Poetria

    Region bounded by a line and a parabola (polar coordinates)

    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

    I have a quadratic regression model ##y = ax^2 + bx + c + \text{noise}##. I also have a prior distribution ##p(a,b,c) = p(a)p(b)p(c)##. What I need to calculate is the likelihood of the data given solely the extremum of the parabola (in my case a maximum) ##x_{max} = M = -\frac{b}{2a}##. What I...
  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 [ASK] Reflected Parabola

    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

    Projectile motion and parabola

    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

    Projectile question (basic level)

    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

    Movement along a parabola

    I have this problem to solve until tomorrow: A point moves along the parabola r*cos^2(θ/2) = p/2, p > 0, in the direction that θ increases. At the time t=0, the point is on the verge of the parabola. The velocity is v = k*r, k>0. What is the equation of movement, the radial acceleration and...
  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

    Projectiles launched at complementary angles

    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?

    I know the difference between the two, but I was wondering if parabolas ever became so steep that they turned back into an elliptical shape.
  45. I

    MHB Where Should I Begin with Parabolas?

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

    I 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

    How to find a quadratic function from a table of values?

    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...

    I Solving Equations of the Parabola

    How can I solve the equation of a parabola when the focus and Latusrectum are given?
  49. Theia

    MHB Normals of parabola - part 2

    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.