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. J

    MHB Finding the equation of parabola

    Greetings, I‘m trying to analyze the given of directrix x = -1, axis y = 2, and latus rectum as 2 I believe there‘s two possibility equations. I‘m not sure for finding the vertex since I got between ( -.5, 0) and origin itself.
  2. MarkFL

    MHB Find Equation of Parabola Given Focus & Directrix - Jose's Q&A

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  3. MarkFL

    MHB Robin's question at Yahoo Answers regarding the osculating circle of a parabola

    Here is the question: I have posted a link there to this thread so the Op can view my work.
  4. Saitama

    MHB Normals from a point to a parabola

    Problem: STATEMENT-1: Through (h,h+1), there cannot be more than one normal to the parabola $y^2=4x$, if $h<2$. STATEMENT-2: The point (h,h+1) lies outside the parabola for all $h\neq 1$. A)Statement-1 is True, Statement-2 is True; Statement 2 is a correct explanation for Statement-1...
  5. J

    How do I find the equation of this parabola?

    How do I find the equation of this parabola?? I am given a parabola with a "way line" l: x+y+2 = 0 and focus point F = (1;1). How do I find it's equation? I know I am suppost to shift it in some way and maybe mirror it, not sure how though. Need some help. The equation of a parabole is y^2...
  6. M

    Finding closest point from a parabola to a plane

    Homework Statement . Find the point in the parabola ##y^2=x##, ##z=0## closest to the plane ##z=x+2y+8## The attempt at a solution. I've solved some problems where I had to find the closest point from a given surface to another point on the space. In this case, I have to find the...
  7. U

    What is the minimum distance between a tangent and normal of a parabola?

    Tangent and normal parabola Homework Statement P is a point 't' on the parabola y^2=4ax and PQ is a focal chord. PT is a tangent at P and QN is a normal at Q. The minimum distance between PT and QN is equal to Homework Equations The Attempt at a Solution I think minimum distance will...
  8. MarkFL

    MHB Khegan McLane's Math Problem: Rectangle Inscribed Between Parabola & X-Axis

    Here is the question: I have posted a link there to this thread so the OP can see my work.
  9. H

    Finding the length of an arc of a parabola

    Homework Statement y^2 = x from (0,0) to (1,1) Homework Equations L = ∫√(1+[g'(y)]^2) dy The Attempt at a Solution So this is a problem in my textbook that has been bothering me because I can't seem to come up with the same answer. 1. [bounds 0 to 1] 1/2∫ sec^3θ was obtained...
  10. MarkFL

    MHB Find Parabola Given Focus & Directrix - Help Needed

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  11. J

    Find Centroid of Parabola Bounded by x Axis

    Homework Statement y = 16 -x^2 find centroid bounded by x axis Homework Equations x = (1/A) ∫ x(f(x)) dx and (1/A) (1/2)(f(x))^2 dx = y The Attempt at a Solution I just applied it. It is a weird because I would of thought that x would of been at 0. But I didn't get that I x =...
  12. J

    MHB Parabola standard form of equation at x = -5

    Find the standard form of the equation of the parabola with the given characteristics and vertex at the origin. Passes through the point (-5, 1/8); vertical axis. I know that there is no focus of the parabola or equation given for this problem, so how would I solve this problem? Is the correct...
  13. J

    MHB Parabola standard form of equation at x = -1

    Find the standard form of the equation of the parabola with the given characteristics and vertex at the origin. Passes through the point (-1, 1/8); vertical axis. I know that there is no focus of the parabola or equation given for this problem, so how would I solve this problem? Is the correct...
  14. C

    Making a parabola with vacuum?

    Greetings, I am thinking of ways to make a parabolic dish. If you apply vacuum to one side of a flat circular elastic sheet, will it make a parabolic or spherical shape? Thanks
  15. T

    Simple parabola solving problem

    Hi friends I am sticking in solving a parabola equation. Please help me in solving this issue. The problem is as follows - I have given three points which are as follows - These points are lying on a parabola whose equation is Y2 = 4AX If these points are lying on this...
  16. P

    Find the point(s) on the parabola x= y^2-8y+18 closest to (-2,4)

    Homework Statement Find the point(s) on the parabola x= y^2-8y+18 closest to (-2,4) Homework Equations The Attempt at a Solution Using the distance formula I found that d^2 = y^4 - 16y^3 + 105y^2 - 328y + 416 after differentiating I have 2d' = 4y^3 - 48y^2 + 210y -328 I'm not...
  17. B

    Parabola problem about the sag of a wire

    Homework Statement Determine a function for wire between two poles if the arc of the wire is parabola. For every 75m there is 1m of sag. Homework Equations y = a(x-p)^2 + q The Attempt at a Solution I think that you need to make the lowest point of the wire the origin and hence...
  18. T

    Lagrange equations of a spinning parabola

    Homework Statement Consider a bead of mass m sliding without friction on a wire that is bent in the shape of a parabola and is being spun with constant angular velocity ω about its vertical axis. Use cylindrical polar coordinates and let the equation of the parabola be ##z = kρ^{2}##. Write...
  19. Barioth

    MHB Acceleration ball rolling on a parabola

    Hi, I'm trying to find the Acceleration of a ball rolling down on a parabola. If I could find it, then I could integrate it twice and find it's parametric equation given by the time. How could I find this? I tried a few things, but nothing that made any sense. If someone could give me a hint...
  20. D

    Finding a connection track to two railroads (circle and parabola)

    There are two railroads, represented by the curves y=(1/2)x2+7, and x2+y2=1. I am supposed to find a straight line connection so that a atrain can get from one curve to the other. I have drawn a simple diagram to help me picture things: http://i50.tinypic.com/2saeccx.png where (a,b) and (c,d)...
  21. S

    MHB What is the limit of the y-intercept as P approaches O on a given parabola?

    We are given the parabola $y \,=\,ax^2$ . . It opens upward, is symmetric to the y-axis, with vertex at the origin $O$. Select any point $P(p,ap^2)$ on the parabola. Construct the perpendicular bisector of $OP$ . . and consider its $y$-intercept, $b.$ | b|...
  22. E

    Closest approach of a parabola to a point, using lagrange multipliers

    Advanced Calculus of Several Variables, Edwards, problem II.4.1: Find the shortest distance from the point (1, 0) to a point of the parabola y^{2} = 4x. This is the Lagrange multipliers chapter. There might be another way to solve this, but the only way I'm interested in right now is the...
  23. I

    Finding the area of a disk divided by a parabola function.

    Homework Statement The parabola y=1/2 x^2 divides the disk x^2+y^2 <or= to 8 into two equal parts. Find the area of both parts. Homework Equations The Attempt at a Solution I have no idea of how to go about solving this. We haven't done any application problems until now and when...
  24. C

    Finding Slope of a Tangent Line to a Parabola

    Homework Statement I've got the equation of a parabola y=2x^2-4x+1 with point (-1,7) and a tangent line running through it the point. I'm supposed to find the equation of the line. Simultaneously solve this equation with that of the parabola, place the results in form ax^2+bx+c, and find the...
  25. C

    Vertex form of parabola; why x-h, not x+h?

    Hello. The vertex form is y= a(x-h)^2+k, in general. Could it also be defined as y= a(x+h)^2+k? I am wondering about that minus sign. I see no particular use of it. Is it there because of tradition or am i missing something?
  26. B

    Conics/Rewriting Parabola into Standard Form

    25x^2-10x-200y-119=0 Should i divide both sides by 25 and then put y and constant to right side of equation and then complete the square?
  27. A

    Change orbit from circle to parabola

    1(a) Homework Statement A spaceship travels in a circular orbit around a planet. It applies a sudden thrust and increases its speed by a factor f . If the goal is to change the orbit from a circle to a parabola, what should f be if the thrust points in the tangential direction? 1(b) Is...
  28. S

    MHB Reflecting Ray from Concave Mirror: Parabola Focus

    A ray of light is coming along the line $y=b$,($b>0$), from the positive direction of the x-axis and strikes a concave mirror whose intersection with the $x-y$ plane is the parabola $y^2 = 4ax$,($a>0$). Find the equation of the reflected ray and show that it passes through the focus of the parabola.
  29. D

    MHB Solving Real Number Variables & Parabola Equations

    This question was asked in my exam and I could not answer it. I would like to know how it can be solved. If $l$ and $m$ are variable real numbers such that $5l^2+6m^2-4lm+3l=0$, then a variable line $lx+my=1$ always touches a fixed parabola, whose axis is parallel to the x-axis. (a) Find the...
  30. M

    Hyperbolic cosine looks like a parabola

    Hello, I wanted to know why the graph of the hyperbolic cosine function (1/2(e^x)+1/2(e^-x)) looks like a parabola. Is there any reason for this? I suppose the individual exponential functions both go to infinity in a symmetric way... but I wanted a better reason :). Thanks, Mathguy
  31. W

    Projectile motion of half a parabola

    1. I have a ball that is thrown off a 3m high cliff at 20 m/s horizontally, what is the displacement on the X axis, Vf, and tair? (assuming no air resistance) Homework Equations d=Vi*t+(1/2a*t2 The Attempt at a Solution When I tried to solve this problem I couldn't really find an...
  32. C

    MHB Finding the equation of the parabola

    Hello guys, please help me, knowing that the parabola passes through the points A(0,1), B(-1,-2) e C(-2,7). How can i find the equation?
  33. R

    Solving Basketball Parabola Puzzle: Initial Velocity at 35°

    Homework Statement A basketball player is standing 9.5 m from the basket, which is at a height of 3.1m. she throws the ball from a initial height of 2m at an angle of 35 degrees above the horizontal. The ball goes through the basket. Determine intial velocity. PS Assume the max height of the...
  34. U

    Parabola Three Normals Homework: Solving for Coordinates and Equations

    Homework Statement Three normals are drawn from the point (14,7) to the curve \large y^{2} -16x-8y=0. Find the co-ordinates of the feet of the normals. Homework Equations Converting the equation of parabola in the form of a perfect square I get \large (y-4)^{2}=16(x+1) The Attempt at a...
  35. P

    Particle moving along a parabola

    Can someone please check that everything I have done so far is correct? A particle movies along a parabola with the displacement equation s = 0.5t2. (a graph is shown) ::::Part I:::: Suppose x-component is s = Ct i) indicate direction of velocity vector and acceleration at point R (arbitrary...
  36. mesa

    Need help rotating a parabola on cartesian coordinate system

    Okay I need to rotate a parabola on a cartesian coordinate system, y=x^2 by 90 degrees about the origin (either direction) without using piecewise, or inverse functions. Basically I am trying to use translations and deformations to accomplish this. Anyone thoughts?
  37. R

    How can I rotate a parabola.

    How can I "rotate" a parabola. I want to take an image and make it kind of wavy to distort it (like in a capcha form the letters are distorted sometimes with a wave) . So i figure that I could translate the pixels along a parabola. So the function where y=X^{2} would make the image curve...
  38. T

    Finding the equation of a parabola

    Homework Statement Alright so I wasn't sure if I should put this in the Calculus section, but just in case that I would need derivatives to solve this I decided I would put this thread here. http://img99.imageshack.us/img99/9351/capturepf.png The Attempt at a Solution I tried finding...
  39. D

    How to calculate the height of a parabola, given base width and arc length?

    I have a parabola centered at x=0, equation: y = a*x^2 + c, where a is always negative and c always positive. I need to find a way to calculate a and c, if i know: the arc length above the x axis, and the base width, knowing the base width i also know the x-axis intersections x1,2 =...
  40. Y

    Parabola problem - some data given.

    Homework Statement Essentially there is a power line which must be strung between posts on two mountains which are 1669.602m apart. The first mountain is 5m higher than the second. To overcome tension the wire is strung with a sag, 1m of sag for every 75m, or part thereof. This sag is...
  41. S

    Parabola Find a the value of a

    The point P(2, 8) lies on the parabola C with equation y2=4ax. Find a the value of a, b an equation of the tangent to C at P. the value of a is 8, so y^2 = 32x when finding the tangent y = 4\sqrt{2} x^{\frac{1}{2}} so at P \frac{dy}{dx} = \frac{2\sqrt{2}}{\sqrt{2}} so the...
  42. E

    Displacement with a velocity time graph that is a parabola

    Homework Statement How would you find the displacement of a velocity/time graph when it is in the shape of a parabola? Homework Equations I know Velocity x time = displacement. The Attempt at a Solution I know that velocity x time = displacement, but how do you find the area under...
  43. H

    Is there a formula for this parabola?

    Given three arbitrary points on a coordinate system, is there a way to derive an equation that forms the single parabola that passes through all three points? I guess firstly you would have to prove that given three points, only a single parabola passes through all three, but judging by the...
  44. H

    MHB Finding equation of parabola with focus and directrix

    So I found the equation for the perpendicular line to the directrix in order to find the vertex, which I got the line y = x that is perpendicular to the directrix, then solved the system of equations to find the common intersection point, which was (1,1). I used the midpoint formula from the...
  45. sankalpmittal

    Projectile Equation - Parabola

    Here's a equation of a projectile thrown in a parabolic path : y=xtanθ - gx2/2u2cos2θ where x is the corresponding x - coordinate or the range of a projectile at a point , y is the corresponding y - coordinate or the height of projectile at point, θ is the angle made with the horizontal...
  46. I

    Rotation of Axes showing parabola

    I was given the equation: 2sqrt(2)(x+y)^2=7x+9y I need to then: a)Use rotation of axes to show that the following equation represents a parabola b) Find the XY- and xy- coordinates of the vertex and focus c) Find the equation of the directrix in XY- and xy-coordinates Formulas...
  47. X

    Nonstandard Concave Mirror Optics, beyond the parabola.

    Why is the parabola the most bent, concave, effective mirror? What are some properties of a third order mirror (absolute value of x^3)? There is no uniform focal point. Shouldn't shining a light beam along a normal to the x-axis reflect off of the function multiple times? Then take this to the...
  48. H

    Significance of parabola in a line integral?

    If $${\tmmathbf{r}}$$ and $${\tmmathbf{s}}$$ are piecewise smooth paths, which have the same graph, then they are said to be equivalent paths. They either trace out a set of points in the same direction, or in the opposite direction. If they trace out a curve $${C}$$ in the same...
  49. B

    Find the values of m such that the line does not touch the parabola

    Homework Statement for what values of m does the line with equation y = mx - 12 not intersect or touch the parabola with equation y = 2x^2-x-10. Please show working out & explain. Homework Equations The Attempt at a Solution This question was asked on openstudy and there...
  50. J

    Focus of parabola, checking my answer.

    I just had this question in an online test I took, and first I had 16ft, but for some unknown reason I chose to change it to 64ft right before I submitted the test, and of course I got it wrong. So I just want to confirm the answer is indeed 16ft.Homework Statement A parabolic antenna has a...
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