What is Hyperbola: Definition and 108 Discussions

In mathematics, a hyperbola (listen) (adjective form hyperbolic, listen) (plural hyperbolas, or hyperbolae (listen)) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. (The other conic sections are the parabola and the ellipse. A circle is a special case of an ellipse.) If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.
Hyperbolas arise in many ways:

as the curve representing the function




{\displaystyle y(x)=1/x}
in the Cartesian plane,
as the path followed by the shadow of the tip of a sundial,
as the shape of an open orbit (as distinct from a closed elliptical orbit), such as the orbit of a spacecraft during a gravity assisted swing-by of a planet or, more generally, any spacecraft exceeding the escape velocity of the nearest planet,
as the path of a single-apparition comet (one travelling too fast ever to return to the solar system),
as the scattering trajectory of a subatomic particle (acted on by repulsive instead of attractive forces but the principle is the same),
in radio navigation, when the difference between distances to two points, but not the distances themselves, can be determined,and so on.
Each branch of the hyperbola has two arms which become straighter (lower curvature) further out from the center of the hyperbola. Diagonally opposite arms, one from each branch, tend in the limit to a common line, called the asymptote of those two arms. So there are two asymptotes, whose intersection is at the center of symmetry of the hyperbola, which can be thought of as the mirror point about which each branch reflects to form the other branch. In the case of the curve




{\displaystyle y(x)=1/x}
the asymptotes are the two coordinate axes.Hyperbolas share many of the ellipses' analytical properties such as eccentricity, focus, and directrix. Typically the correspondence can be made with nothing more than a change of sign in some term. Many other mathematical objects have their origin in the hyperbola, such as hyperbolic paraboloids (saddle surfaces), hyperboloids ("wastebaskets"), hyperbolic geometry (Lobachevsky's celebrated non-Euclidean geometry), hyperbolic functions (sinh, cosh, tanh, etc.), and gyrovector spaces (a geometry proposed for use in both relativity and quantum mechanics which is not Euclidean).

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  1. snoopies622

    B What is the general transformation formula for uniform proper acceleration?

    I'm not perfectly clear about how Rindler coordinates work. Here's what i do understand: Suppose I'm in an inertial reference frame and i define the location of the events around me with (x,t), (ignoring the y and z directions here) and a spaceship approaches me from afar and then flies away...
  2. Sciencemaster

    B How to find the infinitesimal coordinate transform along a hyperbola?

    I've been told that the infinitesimal change in coordinates x and y as you rotate along a hyperbola that fits the equation b(dy)^2-a(dx)^2=r takes the form δx=bwy and δy=awx, where w is a function of the angle of rotation (I'm pretty sure it's something like sinh(theta) but it wasn't clarified...
  3. Vividly

    I Question about Inverse Derivative Hyperbola function

    Im confused about a certain part of solving an equation. So I used the hyerbola formula to find the answer but I think I did the math wrong. X^2-y^2=c^2 X=1 Y= (2x^5-1)^2 I did the calculations as you can see in the picture but I know I messed up on the square root part. When you square one...
  4. ElectronicTeaCup

    Intersection of a tangent of a hyperbola with asymptotes

    Summary:: Question: Show that the segment of a tangent to a hyperbola which lies between the asymptotes is bisected at the point of tangency. From what I understand of the solution, I should be getting two values of x for the intersection that should be equivalent but with different signs...
  5. G

    Different types of hyperbolas and their properties

    I know the hyperbola of the form x^2/a^2-y^2/b^2=1 and xy=c; but coming across this question I'm put in a dilemma of how to proceed with calculating anything of it - say eccentricity or latus rectum or transverse axis as said. How to generalize a hyperbola (but i don't want a complex derivation...
  6. 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...
  7. K

    Closest approach from initial velocity and impact parameter

    Here were my assumptions: Energy and angular momentum are both conserved because the only force acting here is a central force. The initial angular momentum of this particle is ##L = mv_0b## and we can treat E as a constant in the homework equation given above. I solved for the KE (1/2 mv^2) in...
  8. D

    I How to calculate the equation of a Hyperbola

    How do you calculate the equation of a hyperbola, knowing only that the y intercept is (0,y) and the area bounded at x=0 is 'A'. It stands to reason that this can be calculated, but I can't find a tutorial or something similar online to help me answer this. This isn't homework, I'm just trying...
  9. 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...
  10. C

    Triangle and hyperbola

    Homework Statement Points E and F are the focuses of the hyperbola and point X are on the hyperbola. Determine the size of the main and minor half-axes of the hyperbola. Homework Equations x2 = e2 - f2 x = 8 The Attempt at a Solution I think that eccentricity is 4 units (x/2). But I don’t...
  11. barryj

    Formation of a hyperbola using a plane cutting a doubole con

    Homework Statement My question. When studying conics, the parabola circle, and ellipse can be easily see by passing a plane through a double cone. The hyperbola is generated when the plane is passed through the double cone where it passes through the top and the bottom cone. My question is...
  12. Adgorn

    Polar equation of a hyperbola with one focus at the origin

    Homework Statement Hello everyone, I have an assignment (Spivak's Calculus) to show that the polar equation of a hyperbola with the right focus in the origin is ##r=\frac {±\Lambda} {1+εcos(\theta)}##, but the equation I reached was slightly yet somewhat disturbingly different, and I'm not sure...
  13. D

    I Explanation for the hyperbolic array of light rays.

    Hello, Recently, a solar power tower plant was founded next to where I work. Since it's the tallest object in the area, it's quite hard to miss it. But apart from that, every morning the reflected light is arranged in a hyperbolic- like way, as you can see in the picture. Does anyone have a...
  14. L

    B How can I calculate this hyperbola's equation?

    Hello. I am currently trying to calculate the equation of a hyperbola, which I have little experience with. The hyperbola has a "sphere radius" of 153mm and a "hyperbolic factor" of 21500. I haven't been able to find anything online about what these mean and am lost. The parameters where given...
  15. frostysh

    Hyperbola and the Line in the Cartesian coords.

    Homework Statement We have the hyperbola, the focal stuff of which is on the Abscissa axis. $$x^2 - 2y^2 = 4 $$, and we have a line $$3x - 4y = 2$$, and we need to understand if this two crazy stuff will intersect, or be tangent, or nothing like the previous one. Homework Equations I don't...
  16. A

    Rectangular hyperbola, chord, origin

    Homework Statement Prove that the chord joining the points P(cp, c/p) and Q(cq, c/q) on the rectangular hyperbola xy = c^2 has the equation x + pqy = c(p + q) The points P, Q, R are given on the rectangular hyperbola xy = c^2 . prove that (a) if PQ and PR are equally inclined to the axes of...
  17. mertcan

    I Uniform acceleration and hyperbola

    hi, There is a situation that if t^2-x^2=constant in minkowski space we have uniform acceleration(hyperbola). But, when I look at the derivation of this circumstance, the only thing I have found is: if there is a gravitational field uniformly, then we have uniform acceleration. But I can not...
  18. parshyaa

    I Hyperbola: Why Two Branches Facing Opposite Directions?

    Why hyperbola has two branches facing in opposite direction.
  19. parshyaa

    I Hyperbola: Definition & Math Understanding

    Circle: its a locus of a point which moves such that its distance from a fixed point is constant Ellipse: its a locus of a point which moves such that its distance from two fixed point is constant. These definition makes me understand How scientist/mathematican have invented these conics...
  20. A

    Hyperbola and dot product

    Homework Statement A point P moves so that its distances from A(a, 0), A'(-a, 0), B(b, 0) B'(-b, 0) are related by the equation AP.PA'=BP.PB'. Show that the locus of P is a hyperbola and find the equations of its asymptotes. Homework Equations The Attempt at a Solution AP.PA' =...
  21. D

    Hyperbola: define epicenter word problem

    Homework Statement Hello! Here is the word problem that should be solved based on hyperbola equation (exercise from hyporbola topic): The P-waves (\P" stands for Primary) of an earthquake in Sasquatchia travel at 6 kilometers per second.10 Station A records the waves rst. Then Station B, which...
  22. A

    Hyperbola or ellipse problem

    1. Homework Statement The following question is posed within a section of my A level maths book titled "The Hyperbola" A set of points is such that each point is three times as far from the y-axis as it is from the point (4,0). Find the equation of the locus of P and sketch the locus 2...
  23. Odious Suspect

    When to introduce absolute value in hyperbola expression?

    We begin with this definition of a hyperbola. \left(\overline{F_1 P}-\overline{F_2 P}=2 a\right)\land a>0 Perform a few basic algebraic manipulations. \sqrt{(c+x)^2+y^2}-\sqrt{(x-c)^2+y^2}=2 a \sqrt{(c+x)^2+y^2}=2 a+\sqrt{(x-c)^2+y^2} (c+x)^2+y^2=4 a^2+4 a \sqrt{(x-c)^2+y^2}+(x-c)^2+y^2...
  24. astrololo

    Finding the focus points of a hyperbola

    Homework Statement So I have the following hyperbola x^2/4 - y^2/4 = -1 I need to find the focus points of this hyperbola. What is some analytical way to do this ? Thank yoU! Homework Equations I don't know... The Attempt at a Solution I need some analytical way to be able to do this. Can...
  25. I

    Hyperbola Transformations in Clapeyron-Mendeleev Coordinates

    Back in 10th degree, I have learned that in Clapeyron-Mendeleev coordinates ( eq: p-V) , an Isotherm transformation of an ideal gas ( with constant mass throughout the transformation ) is represented with an arc of an hyperbola. Now, I have learned that hyperbola equation is : x2 / a2 - y2/b2...
  26. A

    Confocal ellipse and hyperbola

    If a hyperbola passes through the focii of the ellipse x^2/25 +y^2/16 =1 and its transverse and conjugate axes coincide respectively with major and minor axes of the ellipse, and if the product of eccentricities of hyperbola and ellipse is 1, find the equation and focus of the hyperbola
  27. Quarlep

    Hyperbola equation

    I am studying leonard susskind lectures and there I saw a hyperbola equation I didnt understand that equation what it means coshw or sinhw or total equation(In 49 minute) what's the "h"in there I need help (you can look also 52 min)
  28. N

    Finding the equation to hyperbola

    Homework Statement [/B] Homework Equations General Hyperbola form: x^2/a^2 - y^2/b^2 = 1 or y^2/a^2 - x^2/b^2 = 1[/B] The Attempt at a Solution I am confused by this because I think I am close to the question but I have something fundamentally wrong. What I know is since it's in a...
  29. D

    Circular interference from the equation of a hyperbola

    Hi all, I had a question that I can't seem to find an answer too. I was hoping people could point me in the right direction, or let me know if there is an "easy" method. It has to do with the classic example of two stones in water producing constructive and destructive interference...
  30. bananabandana

    Cartesian form of a Hyperbola

    Homework Statement Why is it necessarily true that for a hyperbola, the focus length, ##f ## has got to be greater than the semi-major axis , ## a## - ## f >a ## ? Homework Equations - The Attempt at a Solution I needed to derive the cartesian equation of a hyperbola with centre at ##...
  31. davidbenari

    Why is the hyperbola round at the bottom?

    My question is somewhat dumb but for some reason I haven't been able to come up with an answer. Why is the hyperbola round at the bottom? Namely, I'm thinking of any of these two equations x^2-y^2=1 or y^2-x^2=1 These two behave like a line as you approach infinity but then becomes round at...
  32. M

    Please help to find the equation for this hyperbola problem

    Homework Statement I would really appreciate if someone can help me understand how my teacher came up with the answer for this find my equation hyperbola problem. I have enclosed a copy of the problem and the solution provided by my teacher. Thanks. Homework Equations The...
  33. Y

    Finding a Hyperbola w/ 2 given points

    Homework Statement Find the equation for a hyperbola centered at the origin with points (-10,3pi/2) and (2,pi/2) Homework Equations x^2/a^2 -y^2/b^2=1 or y^2/a^2 - x^2/b^2 = 1 * r=ke/(1±ecos(theta)) *cos can be replaced with sin and the ± is either a plus or a minus depending...
  34. S

    Where is the right triangle in sides a, b, c, for a hyperbola?

    An ellipse has some model standard form values, a, b, and c which are easily enough to identify from the graph and parts of the graph related to the ellipse's graph. Seeing the right triangle relating a, b, and c, is easy enough. The Pythagorean Theorem is used to relate these three values...
  35. T

    Hyperbola Activity: Extending w/ Foci, Assymptotes & Point A

    I need to extend this activity somehow, but I forgot this stuff already? I learned this a long time ago, I think this activity is too simple so can someone tell me how to find the foci, assymptotes, etc, and what the "point a" is for...
  36. barryj

    How Is a Hyperbola Formed from Conic Sections?

    Conic sections are formed when a plane cuts a double cone, i.e. two cones placed tip to tip along the same axis. A circle is when the plane is perpendicular to the axis, an ellipes when the plane is slightly canted, a parabola when the plane is EXACTLY parallel to the edge of the cones so that...
  37. N

    Sketching a Hyperbola

    Sketching the graph of xz=4 Z=4/x Now this is not in the form of a hyperbola however it is indeed a hyperbola I get this by taking x to 0 and infinity My question is how to put it in the standard form of a hyperbola to find the equations of the aysmptope
  38. E

    MHB The Reflection Property of a Hyperbola

    Hello, Does anyone have a reference to a proof of the reflective property of a hyperbola? I need a proof that uses the geometric definition of a hyperbola as the locus of points $X$ such that $|XF_1-XF_2|=2a$ for some fixed points $F_1$ and $F_2$ and a positive constant $a$. The proof may also...
  39. T

    Finding the Asymptotes of a Hyperbola

    Given a hyperbola of the form: r=\frac{A}{1+Bsinθ+Dcosθ} what are the polar equations for the asymptotes?
  40. MarkFL

    MHB Finding the Line Normal to a Hyperbola: Domenic's Q&A at Yahoo Answers

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

    Hyperbola Fermat, Geometric Infinite Sum

    Hello everybody, I'm trying to understand some steps in the evolution of calculus, and in a .pdf found in the internet I read the document: http://www.ugr.es/~mmartins/old_web/Docencia/Old/Docencia-Matematicas/Historia_de_la_matematica/clase_3-web.pdf [Broken], in pags. 14-15. I want to solve...
  42. V

    Find perpendicular line to hyperbola without calculus

    Homework Statement What is the equation of the line perpendicular to ## x^2-y^2=1 ## at the point ## (2,\sqrt{3}) ##? Homework Equations The Attempt at a Solution I would automatically know what to do if I could use calculus, but apparently you can solve this without it. I would also...
  43. M

    Finding Polar Equation for hyperbola with Graph xy=16

    Homework Statement Find a polar equation with the graph as xy=16 Homework Equations r = ed/(1+- cos\theta)? I'm not really sure at all. The Attempt at a Solution I also tried using x = rcos\theta and y = r sin\theta but I can't get anything. I know it's a hyperbola.
  44. P

    Kruskal Diagram Hyperbola Function

    hi If I understand it correctly, the hyperbolas in the kruskal diagram define locations with the same space time. Now my question is, how can I make a function out of the hyperbola solved for spacetime on one side and gravity/metric tensor on the other side? Thank you very much for an answer...
  45. S

    Finding the equation of a hyperbola given certain conditions

    Homework Statement Find the equation of the hyperbola whose transverse axis is x = 3 and goes through: The vertices of 2x^2 + y^2 - 28x + 8y + 108 = 0 and the center of x^2 + y^2 - 6x + 4y + 3 = 0. Homework Equations (x - h)^2/a^2 - (y - k)^2/b^2 = 1 The Attempt at a Solution so far I...
  46. Z

    Solving for the Height of a Rectangular Hyperbola

    Homework Statement An arch is the shape of a hyperbola. IF it s 300m wide at its base and has a maximum height of 100m, how high is the arch 30m from the end ? Note: this is a rectangular hyperbola. Homework Equations (y-h)^2 - x^2 = a The Attempt at a Solution I determined the...
  47. J

    Is a cone the degenerate of a 4 dimensional hyperbola?

    Is a cone a the degenerate of a 4 dimensional hyperbola? I only ask because I think it is and I am not sure. I am trying to get better at higher dimensional visualizations. My analogy being that a point is the degenerate of a 3 dimensional cone. With that logic wouldn't that make a cone...
  48. U

    Find the equation of hyperbola

    Homework Statement From the point (2√2,1) a pair of tangents are drawn to \dfrac{x^2}{a^2}-\dfrac{y^2}{b^2} = 1, which intersect the coordinate axes in concyclic points . If one of the tangents is inclined at an angle of tan^{-1}\frac{1}{√2} with the transverse axis of the hyperbola , then...
  49. D

    Isosceles hyperbola locus problem

    Hello all, I would like beforehand to inform you that the translation of the following geometric problem is not very good and consequently you will have to use your mathematical intuition just a little bit. I encountered it while giving admission exams in a Mathematics department to pursue a...
  50. M

    Finding Integral of a Divided Function (Hyperbola?)

    Homework Statement Integrate between y=0 and y=-0.5: ∫((y+0.5)/(0.5-y)) dy Homework Equations Can you please show me how to integrate it...Then I will be able to take it from there and substitute in the appropriate values. The Attempt at a Solution Quotient rule in...