File:Hyperbola_(PSF).svg · Wikimedia Commons · See Wikimedia Commons
hyperbola
thumb|A hyperbola is an open curve with two branches, the intersection of a plane (geometry)|plane with both halves of a double cone. The plane does not have to be parallel to the axis of the cone; the hyperbola will be symmetrical in any case.|alt=The image shows a double cone in which a geometrical plane has sliced off parts of the top and bottom half; the boundary curve of the slice on the cone is the hyperbola. A double cone consists of two cones stacked point-to-point and sharing the same axis of rotation; it may be generated by rotating a line about an axis that passes through a point of
AI overview
A hyperbola is an open curve with two separate branches that forms when a plane cuts through both halves of a double cone at any angle. This shape is important in mathematics and science because it appears naturally in various physical phenomena and serves as a fundamental curve in geometry alongside circles, ellipses, and parabolas.
AI-generated from the Wikipedia summary — may contain errors.
Described at
Equations of Hyperbolas | College Algebra
courses.lumenlearning.com →In analytic geometry a hyperbola is a conic section formed by intersecting a right circular cone with a plane at an angle such that both halves of the cone are intersected. This intersection produces two separate unbounded curves that are mirror images of each other. As with the ellipse, every hyperbola has two axes of symmetry . The transverse axis is a line segment that passes through the center of the hyperbola and has vertices as its endpoints. The foci lie on the line that contains the transverse axis. The conjugate axis is perpendicular to the transverse axis and has the co-vertices as its endpoints. The center of a hyperbola is the midpoint of both the transverse and conjugate axes, where they intersect. Every hyperbola also has two asymptotes that pass through its center. As a hyperbola recedes from the center, its branches approach these asymptotes. The central rectangle of the hyperbola is centered at the origin with sides that pass through each vertex and co-vertex; it is a useful tool for graphing the hyperbola and its asymptotes. To sketch the asymptotes of the hyperbola, simply sketch and extend the diagonals of the central rectangle. Note that the vertices, co-vertices, and foci are related by the equation c2 =a2+b2. When we are given the equation of a hyperbola, we can use this relationship to identify its vertices and foci. How To: Given the equation of a hyperbola in standard form, locate its vertices and foci. 1. Determine whether the transverse axis lies on the x – or y -axis. Notice that a2 is always under the variable with the positive coefficient. So, if you set the other variable equal to zero, you can easily find the intercepts. In the case where the hyperbola is centered at the origin, the intercepts coincide with the vertices. Identify the vertices and foci of the hyperbola with equation y249−x232 =1. Identify the vertices and foci of the hyperbola with equation x29−y225 =1. Just as with ellipses, writing the equation for a hyperbola in standard form allows us to calculate the key features: its center, vertices, co-vertices, foci, asymptotes, and the lengths and positions of the transverse and conjugate axes. Conversely, an equation for a hyperbola can be found given its key features. We begin by finding standard equations for hyperbolas centered at the origin. Then we will turn our attention to finding standard equations for hyperbolas centered at some point other than the origin. Reviewing the standard forms given for hyperbolas centered at (0,0), we see that the vertices, co-vertices, and foci are related by the equation c2 =a2+b2. Note that this equation can also be rewritten as b2 =c2−a2. This relationship is used to write the equation for a hyperbola when given the coordinates of its foci and vertices. What is the standard form equation of the hyperbola that has vertices (±6,0) and foci (±210,0)? The vertices and foci are on the x -axis. Thus, the equation for the hyperbola will have the form x2a2−y2b2 =1. What is the standard form equation of the hyperbola that has vertices (0,±2) and foci (0,±25)? Using the reasoning above, the equations of the asymptotes are y =±ab(x−h)+k. Like hyperbolas centered at the origin, hyperbolas centered at a point (h,k) have vertices, co-vertices, and foci that are related by the equation c2 =a2+b2. We can use this relationship along with the midpoint and distance formulas to find the standard equation of a hyperbola when the vertices and foci are given. The y -coordinates of the vertices and foci are the same, so the transverse axis is parallel to the x -axis. Thus, the equation of the hyperbola will have the form Finally, substitute the values found for h,k,a2, and b2 into the standard form of the equation. What is the standard form equation of the hyperbola that has vertices (1,−2) and (1,8) and foci (1,−10) and (1,16)? As we discussed at the beginning of this section, hyperbolas have real-world applications in many fields, such as astronomy,
~56 min read
Article
62 sectionsContents
- Etymology and history
- Definitions
- As locus of points
- Hyperbola with equation {{math|1=''y'' = ''A''/''x''}}
- By the directrix property
- Proof
- Construction of a directrix
- As plane section of a cone
- Pin and string construction
- Steiner generation of a hyperbola
- Inscribed angles for hyperbolas {{math|1=''y'' = ''a''/(''x'' − ''b'') + ''c''}} and the 3-point-form
- As an affine image of the unit hyperbola {{math|1=''x''<sup>2</sup> − ''y''<sup>2</sup> = 1}}
- Parametric representation
- Vertices
- Implicit representation
- Hyperbola in space
- As an affine image of the hyperbola {{math|1=''y'' = 1/''x''}}
- Tangent construction
- Point construction
- Tangent–asymptotes triangle
- Reciprocation of a circle
- Quadratic equation
- In Cartesian coordinates
- Equation
- Eccentricity
- Asymptotes
- Semi-latus rectum
- Tangent
- Rectangular hyperbola
- Parametric representation with hyperbolic sine/cosine
- Conjugate hyperbola
- In polar coordinates
- Origin at the focus
- Origin at the center
- Eccentricity
- Parametric equations
- Hyperbolic functions
- Properties
- Reflection property
- Midpoints of parallel chords
- Orthogonal tangents – orthoptic
- Pole-polar relation for a hyperbola
- Other properties
- Arc length
- Derived curves
- Elliptic coordinates
- Conic section analysis of the hyperbolic appearance of circles
- Applications
- Sundials
- Multilateration
- Path followed by a particle
- Korteweg–de Vries equation
- Angle trisection
- Efficient portfolio frontier
- Biochemistry
- Hyperbolas as plane sections of quadrics
- See also
- Other conic sections
- Other related topics
- Notes
- References
- External links
thumb|A hyperbola is an open curve with two branches, the intersection of a plane (geometry)|plane with both halves of a double cone. The plane does not have to be parallel to the axis of the cone; the hyperbola will be symmetrical in any case.|alt=The image shows a double cone in which a geometrical plane has sliced off parts of the top and bottom half; the boundary curve of the slice on the cone is the hyperbola. A double cone consists of two cones stacked point-to-point and sharing the same axis of rotation; it may be generated by rotating a line about an axis that passes through a point of the line. thumb|Hyperbola (red): features thumb|A lampshade with a circular rim casts a hyperbola-shaped shadow on a vertical wall. In mathematics, a hyperbola 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.
Besides being a conic section, a hyperbola can arise as the locus of points whose difference of distances to two fixed foci is constant, as a curve for each point of which the rays to two fixed foci are reflections across the tangent line at that point, or as the solution of certain bivariate quadratic equations such as the reciprocal relationship xy = 1. In practical applications, a hyperbola can arise as the path followed by the shadow of the tip of a sundial's gnomon, the shape of an open orbit such as that of a celestial object exceeding the escape velocity of the nearest gravitational body, or the scattering trajectory of a subatomic particle, among others.