Pronunciation: /haɪˈpɜr.bə.lə/ Explain
|Figure 1: Conic section||
A hyperbola is a
a right circular
surface and a plane that intersects both halves of the cone. The equations
most often used for a hyperbola are:
where a is the semi-major axis parallel to the x-axis,
b is the semi-minor axis parallel to the y-axis, and
the point (h,k) is the center of the hyperbola, and
where a is the semi-minor axis parallel to the x-axis,
b is the semi-major axis parallel to the y-axis, and
the point (h,k) is the center of the hyperbola.
Graph of a Hyperbola
Click on the blue points and drag them to change the figure.|
Why does the hyperbola disappear when a or b is zero?
|Manipulative 1 - Hyperbola Created with GeoGebra.||
A hyperbola can be defined as a set of points where the ratio
of the distance from a fixed line called the
directrix and the distance from
the point is equal to the ratio of c to a.
The equation for the directrices of a hyperbola with an east/west opening is
The equation for the directrices of a hyperbola with a north/south opening is
The foci for a hyperbola are the points at
for a hyperbola with an east/west
opening and for a hyperbola
with a north/south opening.
In manipulative 1,
click on the check box marked 'Show directrices'. Click on the blue point on the
hyperbola and drag it. Notice that the ratio remains constant for
a particular hyperbola.
Eccentricity of Hyperbolas
The eccentricity of a hyperbola can be
considered as how far the hyperbola deviates from a circle. The larger
the eccentricity, the flatter the hyperbolic curve. The formula for the
eccentricity of a hyperbolic curve is
Hyperbolas in Nature
A flashlight makes a cone of light. If a flashlight is held close to a wall and
parallel to the wall, the plane of the wall 'cuts' the cone at
in such a way that the 'edges' of the light form a hyperbola.
|Figure 2: Flashlight hyperbola. Click on the image to see a larger version.|
A liquid in a small tube is drawn up the tube by a action known as
capillary action. The liquid forms a hyperbolic curve.
For more information see
in The Internet Encyclopedia of Science.
|Figure 3: Capillary action.|
The path of an object in space relative to a much larger object can
be a hyperbola. The path of the object is a hyperbolic
trajectory if the speed of the smaller object relative to the
larger object is more than escape velocity. In actuality, very few
hyperbolic trajectories are observed in nature. This is because an
object on a hyperbolic trajectory 'breaks free' of the gravitational
field of the larger object (in this case the sun). Because it breaks free, the object does
not return close to the larger object so is rarely observed.
|Figure 4: Hyperboloid of one sheet.||
A hyperboloid is a 3-dimensional figure created by
rotating a hyperbola about a line. Figures 4 and 5 show two types of hyperboloids.
|Figure 5: Hyperboloid of two sheets.|
- McAdams, David E.. All Math Words Dictionary, hyperbola. 2nd Classroom edition 20150108-4799968. pg 92. Life is a Story Problem LLC. January 8, 2015. Buy the book
- McAdams, David E.. Conic Section. allmathwords.org. Life is a Story Problem LLC. 3/12/2009.
Cite this article as:
McAdams, David E. Hyperbola. 3/17/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/h/hyperbola.html.
- All images and manipulatives are by David McAdams unless otherwise stated. All images by David McAdams are Copyright © Life is a Story Problem LLC and are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
- Hyperbolic Conic Section. Unknown.
- Hyperboloid of One Sheet: Svdmolen, https://en.wikipedia.org. Licensed under the terms of the GNU Free Documentation License, Version 1.2 or any later version. https://en.wikipedia.org/wiki/File:HyperboloidOfOneSheet.png.
- Hyperboloid of Two Sheets: Svdmolen, https://en.wikipedia.org. Licensed under the terms of the GNU Free Documentation License, Version 1.2 or any later version.
- Flashlight Making a Hyperbola: Dave Clauson. Used with permission. Do not copy.
3/17/2019: Fixed broken link. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
7/16/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
2/8/2010: Added "References". (McAdams, David E.)
1/7/2010: Added "References". (McAdams, David E.)
12/12/2008: Added hyperboloid. (McAdams, David E.)
12/10/2008: Corrected equations. Added hyperbolas in nature (McAdams, David E.)
11/26/2008: Change equations to images. (McAdams, David E.)
8/8/2008: Added paragraph on eccentricity. Added http://www.allmathwords.org/conicsection.html to More Information. Added manipulative 1. Converted equation to hot_eqn (McAdams, David E.)
7/1/2008: Initial version. (McAdams, David E.)