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. 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.
  2. Theia

    MHB What are the normals of a parabola passing through a given point?

    Find all the normals of function y = 2x^2 + 4x + \tfrac{7}{4} which goes through the point \left( 3, \tfrac{15}{2} \right).
  3. S

    I How directrix for other than parabola?

    I do not understand this: An ellipse has a directrix? A parabola, for sure. This is part of the definition and we can derive the equation of a parabola using the meaning of Directrix and the meaning of Parabola. How does a directrix work for an ellipse?
  4. S

    Tangent line and normal on a parabola

    Homework Statement If the normal at P(ap^2 ,2ap) to the parabola y^2 = 4ax meets the curve again at Q(aq^2, 2aq), show that p^2 +pq+2=0 Homework Equations Point-slope form The Attempt at a Solution I tried putting y=2aq and x=aq^2 but I can seem to simplify the whole thing other than...
  5. BlueQuark

    Finding the focus of a parabola given an equation

    Homework Statement Alright, so the equation of a parabola is y = 1/4p*x^2, P being either an x or y value, and the other x or y being zero. Let's say that x^2 = 16y. If you divide both sides by 16, you get y = x^2/16, which can be simplified to y = 1/16*x^2. This is in the format of a parabola...
  6. D

    Downward parabola: value of p = 1/a?

    Homework Statement Hello! I am repeating conics, and have stumbled upon this paragraph on purplemath.com " Then a = –1/9. With a being the leading coefficient from the regular quadratic equation y = ax^2 + bx + c, I also know that the value of 1/a is the same as the value of 4p, so 1/(–1/9) =...
  7. karush

    MHB *Questions about a parabola

    The graph of $y=(x-2)^2 - 1$ is given, then they say use this graph for problem 7 Next is 7) $f(x) =\frac{1} {g(x)} $ ? Step one restrictions $y> \ge - 1$ Step two Equation Solution? Step three: Domain:= $\left(-\infty, +\infty\right)$ Vertical Asymptopes at: none What's with the?
  8. D

    Arriving at parabola formula via distance formula

    Homework Statement Hello! Please, help me to get through equations. I can't derive the equation in the way suggested. Here is the definition: So, if vertex is at (h, k) and there is a given point on a parabola at (x; y), then focus is at (h; k + p), and point of a directrix is at (x; k -...
  9. Cosmophile

    Drawing a parabola with different labels

    Hey, all. I'm wanting to type up a few ways to solve a neat Kleppner & Kolenkow problem, and I'd like to put a diagram I've drawn in. Here is what I'd like to insert...
  10. T

    Triangle formed by tangents to a parabola

    Homework Statement The triangle formed by the tangents to the parabola y2=4ax, at the ends of the latus rectum and the double ordinate through it's focus is: 1) equilateral 2) acute angles isosceles 3)right angled isosceles 4) dependent on value of a Homework Equations...
  11. B

    Why is a projectile a parabola not a semicircle

    This sounds like a dumb question. I have come to accept projectiles form parabolas but I need someone to explain why they form this shape
  12. S

    Vertex of Parabola: Find y^2-12=12x Solution

    Homework Statement I'm trying to find the vertex of this parabola: y^2-12=12x The Attempt at a Solution Initially I tried to take the derivative and find the vertex by finding where it the function is zero, but apparently that does not work in this situation. Is it because it is not in...
  13. K

    Integral (area) of a revolved parabola

    Whats's the volume under the revolving parabola y=ax2. R is xmax. $$V=\int_0^R\pi xdx\cdot y=\pi\int_o^R x\cdot Ax^2dx=\pi A\frac{R^4}{4}$$ Volume should be relative to R3. and if i had, for example, y=Ax3 then, according to my calculation i would get relative to R7 and so on.
  14. gracy

    Upside down parabola equation -- needed for a quiz

    http://www.chilimath.com/flash-quiz/algebra/intermediate/oef/odd-even-functions-quiz.html']<[/PLAIN] Mentor Note -- thread moved to HH from the technical math forums, so no HH Template is shown > http://www.chilimath.com/flash-quiz/algebra/intermediate/oef/odd-even-functions-quiz.html In this...
  15. S

    Evaluate Magnitude of Gravitation at the Surface of a Planet

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  16. F

    Exploring Tangents on Parabolas: A Comparison of y^2=4ax and x^2=4ay

    I know a tangent drawn on parabola having equation like y^2=4ax is y=mx+(a/m) which provides c=a/m Then how is it going to turn for equation like x^2=4ay? From my derivation it will be like -c=am^2 when the equation of tangent is y=mx+c. The derivation comes from the following: y=mx+c or...
  17. Jaco Viljoen

    Finding the equation of a parabola with 1 point

    Homework Statement The sketch on the previous page shows the graph of a function f, which is a parabola with vertex R, and the graph of a function g, which is a straight line defined by g(x)=-(1/2)x. The graphs of f and g intersect at P and O(the origin). The function f is defined by...
  18. R

    The orthocentre of the triangle and a parabola

    Homework Statement The orthocentre of the triangle formed by points t1,t2, t3 on the parabola y2 = 4ax is vertex Origin Focus (1,0) Homework Equations NA The Attempt at a Solution The points can be taken anywhere, So orthocentre can be formed anywhere isn't it?
  19. C

    Integrating law of gravity into a parabola?

    Homework Statement Hey all, I've been learning some incredibly INCREDIBLY basic calculus on my own, so please take it easy on my stupidity. So here's what I was wondering. In a 1 dimensional theoretical system, let the acceleration experienced by an object = A, with the signage +/- indicating...
  20. Drakkith

    Equations of two Lines that are Tangent to a Parabola

    Homework Statement [/B] Find equations of both lines through the point (2,-3) that are tangent to the parabola y=x2+x. Homework Equations Slope formula: (Y2-Y1)/(X2-X1) = M The Attempt at a Solution Here's what I think I need to do. First I think I need to find the derivative of the...
  21. N

    Finding the Axis of Symmetry in a Parabola: How to Solve the Problem

    Homework Statement The Attempt at a Solution I would guess it is -b/2a but I have gotten this incorrect. Maybe I have to figure out what the parabola equation is and apply it into the axis of symmetry. Assuming that is correct how would I find that? Thanks[/B]
  22. R

    Finding acceleration of a skier on parabola

    Homework Statement A skier travels with a constant speed of 6 m/s along a parabolic path y= x2/20. Find the acceleration of the skier when he is at (10,5). Neglect the size of skier. Homework Equations dy/dx is slope of parabola. In a straight line if body has distance covered=x, then velocity...
  23. A

    Understanding Parabola: Investigating a Trajectory Lab

    Homework Statement So I am doing a trajectory lab. I take a ball and release it from the ramp that is on the table and then measure the distance. The thing is when I graph the angle of the ramp vs the distance the ball has fallen I get parabola. And the correlation of parabola is 0 Homework...
  24. Shaheen Uqab

    Calculate moment of inertia of a "L" angle bent in a parabola

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  25. B

    Question about concave mirrors

    I read that concave mirrors are part of a sphere, and concave mirrors can also be expressed in a parabola equation, but a parabola equation is expressed as #4py=x^2# and a circle as #x^2+y^2=R^2#. So the two can't be the same right? Can someone please explain this? Thank you in advance :)
  26. 2

    Equation for a tilted parabola in 3D?

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  27. angeli

    Application of quadratic functions to volleyball

    hi! i don't quite know how to start solving for this. i understand the problem and what it's asking for but i have no idea how to start solving for it. In a volleyball game, a player from one team spikes the ball over the net when the ball is 10 feet above the court. The spike drives the ball...
  28. T

    Locus of mid point of parabola and straight line

    Homework Statement A line passing through the point (1,0) cuts the curve Y^2=4x at two points A and B. Find the equation of the locus of the mid-point of AB. Homework EquationsThe Attempt at a Solution the graph is parabola, the solution should also be a parabola with x-intercept at (1,0)...
  29. Z

    Parabola incoming ray not parallel to axis

    If incoming light to a parabola is parallel to the central axis the light is reflected through the focal point and the back to its source. What about light coming in not parallel to the central axis. What path do these rays take?
  30. S

    MHB Find the equation of parabola with vertex (2, 1) and focus (1, -1)

    Hello friends. This is my first post. I am given the following problem: Find the equation of parabola with vertex (2, 1) and focus (1, -1) I have tried to solve this question in this way: Since vertex is mid point of Focus and the point which touches the Directrix. From this, I calculate that...
  31. E

    Find Minimum Value of T^2 in Normal to Parabola y^2 = 4ax at (at^2, 2at)

    Homework Statement If the normal to the parabola y^2 = 4ax at the point (at^2 , 2at) cuts the parabola again at (aT^2, 2aT), then minimum value of T^2 is ans: 8 I got the answer but I don't know why it should be the answero_O? Homework Equations Equation of normal to the parabola in parametric...
  32. I

    MHB Graphing x=t, y=1/(1+t^2), z=t^2 - Is it a Parabola?

    i don't know whether this goes here or under the calculator forum. i want to graph x=t y=1/(1+t^2) and z=t^2 and i keep getting a parabola. is that right because the drawing in my book is like a slanted parabola.
  33. K

    Equation of a Parabola: Can Any Point on the Graph Satisfy the General Equation?

    The general equation of a parabola is: (y - y1) = A(x - x1)^2 A is the scaling factor y1 is the y coordinate of a vertex/point on the parabola x1 is the x coordinate of a vertex/point on the parabola My two questions are: 1. Can (x,y) be any point on the graph ? 2. If so, then if x -...
  34. Greg Bernhardt

    What is a Parabola? A 5 Minute Introduction

    Continue reading...
  35. rakeru

    Two values for m for which a line is tangent to a parabola

    Homework Statement Hi! I am doing some problems to practice for a math competition, and I'm wondering if I did this correctly. I don't really have an answer sheet, so I have no way of knowing whether I'm right. If you would please review it, that would be cool! It reads: There are...
  36. A

    MHB How to Determine the Vertex of a Parabola?

    The general equation of the parabola has the form $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ with $B^2 = 4AC$ what is the coordinate of the vertex of the parabola or in other word how to determine the vertex of a given parabola for example $4x^2 + 4xy + y^2 - 5x + 7y + 11 =0 $ Thanks.
  37. Adjoint

    A particle moving in parabola (Kinematics)

    Homework Statement A particle is moving along a parabola y = x2 so that at any time vx = 3 ms-1. Calculate the magnitude and direction of velocity and acceleration of the particle at the point x = 2/3 m. Homework Equations Kinematics The Attempt at a Solution vx = 3 (given) y = x2 so vy =...
  38. karush

    MHB Volumn of parabola and line with perpendicular cross sections being a square.

    The base of a solid is the region bounded by the parabola y^2=4x, and the line x=2 . Each plane section perpendicular to the x-axis is square. (I assume this means the cross-section of the solid will be square) then we are not revolving but slicing. The volume of the solid is? (the ans is...
  39. V

    Solving Two Parabola Problems: Proving & Finding Values

    I am not able to solve the following problem #1) Prove that the normal to parabola y2=4ax at (am2,-2am) intersects the parabola again at an angle tan-1(m/2) What I am thinking is to solve the equation of parabola and equation of normal y=mx-am-3-2am simultaneously and at that point I will...
  40. S

    Point of intersection between a parabola and a circle

    Homework Statement Sketch the curve C defined parametrically by ##x=t^{2} -2, y=t## Write down the Cartesian equation of the circle with center as the origin and radius ##r##. Show that this circle meets the curve C at points whose parameter ##t## satisfies the equation ##t^{4} -3t^{2}...
  41. adjacent

    Proof: Parabola reflects light to the focus

    Homework Statement The title.Homework Equations ##y=mx+c## $$\frac{dy}{dx}(kx^2)=2kx$$ The Attempt at a Solution https://www.physicsforums.com/attachment.php?attachmentid=69745&stc=1&d=1399976415 I have proved that the angle of reflection will be a° since the angle of incidence=a° The picture...
  42. sheldonrocks97

    Parametric equations for the portion of the parabola y=x^2?

    Homework Statement Find the parametric equations for the portion of the parabola y=x^2 from (-1,1) to (3,9) Homework Equations None that I know of. The Attempt at a Solution Using knowledge of parametric equations I am not sure how to start. My teacher never went over this...
  43. L

    MHB What is the intercept of N when MN is minimal in a parabola y2 = 2px?

    the normal at a point N of the parabole: y2 = 2px to the curve at another point M. calculate the intercept of N when the length of MN is minimal. Answer sqrt (2) p
  44. P

    MHB Finding equation of parabola with 3 points given

    Hello, I am supposed to find the equation of a parabola with the points (8,10)(11,10)(10,20/3). I have tried putting these values into y=ax^2+bx+c, but get different answers each time, like c=-160, which is not right! A step by step explanation would be greatly appreciated, as I am very unsure...
  45. W

    Hamiltonian of a pendulum constrained to move on a parabola

    Homework Statement The point of suspension of a simple pendulum of length l and mass m is constrained to move on a parabola z = ax^2 in the vertical plane. Derive a Hamiltonian governing the motion of the pendulum and its point of suspension. This is a two-dimensional problem...
  46. L

    MHB Finding the point on a parabola closest to the origin

    Find the curve coordinates of the point nearest to P in the curve 2y2 = 5(x+1) P(0,0) Book answer (-1,0) ok (x-0)2+ (y-0)2 = D D = (x)2+ (y)2 and (y)2 = -x2 Now -x2 = 5/2(x+1) But derivating I don't get the answer or ansa
  47. L

    MHB Finding the point on a parabola closest to a given point

    Find the curve coordinates nearest to P a)x2 = y P(3,0) ok to the boox exercise model I must do so (x-3)2+ (y)2 = D then x2-6x +9+y2 = D introducing y into x y-6sqrt(y) +9 + (y)2 = D derivating 1-3/(sqrt(y) +0+2y mcm = sqrt(y) -3 +2y = 0 solving sqrt(y) = -2y+3 Squaring y =...
  48. D

    Circle Inscribed in a Parabola?

    Homework Statement Find the largest circle centered on the positive y-axis which touches the origin and which is above y=x^2 Homework Equations equation of a circle: r^2=(x-a)^2+(y-b)^2 equation of a circle centered on the y-axis: x^2+(y-b)^2=r^2 equation of a parabola: y=x^2 The...
  49. W

    What is the process for finding the unit normal at every point on a parabola?

    I am working on a fluid mechanics problem that has a parabolic gate with equation y = x^2 To solve the problem I need two vectors namely \vec{r} \ and \ \hat{n} . Assuming origin is at x = 0, the vector \vec{r} is a vector corresponding to each point on the parabola. I calculated that to...
  50. T

    MHB Help me cut the area under a parabola in half

    The problem is: Consider the area under the curve f(x)=2x-x2 and above the x axis. Find the equation of the line through the origin that cuts this area into two equal parts.
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