What is hyperbolas: Definition and 19 Discussions

In mathematics, a hyperbola ( (listen); pl. hyperbolas or hyperbolae (listen); adj. hyperbolic (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 reciprocal 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 (or celestial object) exceeding the escape velocity of the nearest planet or other gravitational body,
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. chwala

    Solve the given simultaneous equations that involves hyperbolas

    My approach on this; ##\dfrac{x^2}{4}-\dfrac{y^2}{9}=\dfrac{y^2}{4}-\dfrac{x^2}{9}## ##9x^2-4y^2=9y^2-4x^2## ##13x^2-13y^2=0## ##x^2=y^2## Therefore, on substituting back into equation we shall have; ##\dfrac{x^2}{4}-\dfrac{x^2}{9}=1## ##9x^2-4x^2=36## ##5x^2=36## ##x^2=7.2##...
  2. 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...
  3. T

    Understanding Hyperbolas: When Objects Move Horizontally

    when an object is thrown horizontally ,after some time when the effect of the applied force is less than the effect of gravity then it changes its path and bends towards the Earth and the path is called a parabola ,then in the same sense how can we define hyperbola?
  4. A

    MHB Family of hyperbolas isomorphic to A^2

    This question is typically seen in the beginning of a commutative algebra course or algebraic geometry course. Let $V = \mathcal{Z}(xy-z) \subset \mathbb{A}^3$. Here $\mathcal{Z}$ is the zero locus. Prove that $V$ is isomorphic to $\mathbb{A}^2$ as algebraic sets and provide an explicit...
  5. R

    Hyperbolic geometry - relations between lines, curves, and hyperbolas

    Hi. I studied calculus a while back but am far from a math god. I have been reading around online about hyperbolic geometry in my spare time and had a simple question about the topic. If a straight line in Euclidean geometry is a hyperbola in the hyperbolic plane (do I have that right?)...
  6. jaumzaum

    Second Kepler law for hyperbolas

    A body can describe 3 types of orbits around another (considering only the gravitational force), a elipse, a hyperbole or a parabola. Does the second kepler law (area law) work for hyperboles or parabolas too?
  7. evinda

    MHB Graph of f(x,y): Contour Curves & Hyperbolas

    Hi! Let the function be f(x,y)=x^2-2*y^2,which graph is S:z=f(x,y).Which are the contour curves?Are these hyperbolas? :confused: Thanks in advance!:)
  8. P

    Equation of tangents of hyperbolas

    find the equation of the tangents to the hyperbola H` with equation \frac{x^2}{25} - \frac{y^2}{16} = 1 at the point (1,4) in an earlier part of the equation we had to prove that a tangent to the a hyperbola in the form of \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 is in the form of a^2m^2 =...
  9. S

    Hyperbolas of constant t^2 - x^2

    I know hyberbola in spacetime diagram is that curve-like line below the light cone much like the bottom of a bath tub. But what does "hyperbolas of constant t^2 - x^2" represent? Can anyone point to a spacetime diagram or draw it? Thanks. Happy New Year!
  10. S

    Graph and Differential equations for hyperbolas

    Hello experts! Hope all of you will be fine. I have an equation i.e. xy=c And we all know it is hyperbola. Now I say "graph some of the hyperbolas xy=c". Then kindly tell me how can we extract more than 1 graph from this single equation? And you will write the differential equations for...
  11. Z

    How do I find the equation of a hyperbola given a point and one asymptote?

    Homework Statement 1) A hyperbola goes through the point P(6, 2), and one of its asymptotes is the line r: 2x + 5y = 0. Determine its equation. 2) Prove that a line parallel to one asymptote of a hyperbola interesects it in a single point. Homework Equations The Attempt at a...
  12. D

    General Question Regarding Hyperbolas

    The template doesn't quite fit my question, so sorry for not using it :/ If I know the distance from the focus to the vertex of a hyperbola, is it possible to find the equation of the hyperbola, assuming the center is at (0,0) on a Cartesian plane? If yes, how do you do so? Using typical...
  13. T

    Parabolas and hyperbolas - how do you find them?

    Finding linear equations for initial conditions has never been a problem from me since is easy to see (x1, y1) and (x2, y2) can be plugged into the slope equation. But what if I'm looking for a parabolic or even a hyperbolic line, how do I then find its equation with the initial condition that...
  14. X

    Maximal area of a convex region bounded by hyperbolas

    This problem was suggested by Gokul43201, based on this year's Putnam A2. Suppose that K is a convex set in \mathbb{R}^2 which is contained in the region bounded by the graphs of the hyperbolas xy=1, xy=-1 (so the set is in the inner + shaped region which contains the origin also). What is...
  15. C

    Asymptotes for hyperbolas q2 problem :S

    Does the equation (x+1)^2-4y^2 = 0 have asymptotote?? i graphed it and from the graph it does not look like a hyperbola because it seems to intersect at the point x = -1 y = 0 :frown: Thanks HallsofIvy for helpin me with the previous asymptote hyperbola prob
  16. C

    How do you find the asymptotes of a hyperbola?

    How do u find the asymptotes to hyperbolas?? what are the asymptotes to this equation? -x^2/4 + y^2/8 = 1 i really need help :confused:
  17. V

    Solving Precal Problems: Ellipses & Hyperbolas

    Here are two problems that stumped our entire precal class. And we have a test soon, so I would like to be able to know how to work these type of problems. 1. Write the equation of the hyperbola, x^2 + 4xy + y^2 - 12 = 0, in standard form. Okay, I know the formula needs to be x^2/a^2 -...
  18. H

    Questions about hyperbolas

    Hey, I have a couple of _easy_ questions about hyperbolas, but its been a while since I have worked with them and am not able to look them up in my math book currently...if someone could just get me started in the right direction, I would really appreciate it :) Given the equation of the...
  19. W

    Hyperbolas equivalent of an ellipsis's minor axis?

    This is algebra. I think I missed this part in class. I know a Hyperbola has a conjugate axis, but is this the hyperbolas equivlent of an elipise's minor axis? If so, how do you find the measurement? I'm writing a forumla for a hyprbola in standard form and need to find b^2. Thanks.