hyperbola
hyperbola (hīpûr´bələ), plane curve consisting of all points such that the difference between the distances from any point on the curve to two fixed points (foci) is the same for all points. It is the conic section formed by a plane cutting both nappes of the cone; it thus has two parts, or branches. The center of a hyperbola is the point halfway between its foci. The principal axis is the straight line through the foci. The vertices are the intersection of this axis with the curve. The transverse axis is the line segment joining the two vertices. The latus rectum is the chord through either focus perpendicular to the principal axis. The asymptotes are lines, in the same plane, which the curve approaches as it approaches infinity. An equilateral, or rectangular, hyperbola is one whose asymptotes are perpendicular. A second hyperbola may be drawn whose asymptotes are identical with those of the given hyperbola and whose principal axis is a perpendicular line through the center; the two hyperbolas thus related are called conjugate.
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hyperbola
hy·per·bo·la / hīˈpərbələ/ • n. (pl. bo·las or bo·lae / bəlē/ ) a symmetrical open curve formed by the intersection of a cone with a plane at a smaller angle with its axis than the side of the cone. ∎ Math. the pair of such curves formed by the intersection of a plane with two equal cones on opposites of the same vertex.
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hyperbola
hyperbola Plane curve traced out by a point that moves so that its distance from a fixed point bears a constant ratio, greater than one, to its distance from a fixed straight line. The fixed point is the focus, the ratio is the eccentricity, and the fixed line is the directrix. The curve has two branches and is a conic section. Its standard equation in Cartesian coordinates x and y is x^{2}/a^{2}−y^{2}/b^{2} = 1.
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hyperbola
hyperbola. Conic section formed by the intersection of a plane with both branches of a double cone (two identical cones on either side of the same vertex or pointed top).
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hyperbola
hyperbola
•areola, rubeola
•Viola
•dueller (US dueler), jeweller (US jeweler)
•babbler, dabbler, parabola
•labeller (US labeler)
•dribbler, nibbler, quibbler, scribbler
•libeller (US libeler)
•hobbler, nobbler, squabbler, wobbler
•bubbler
•fumbler, mumbler, rumbler
•burbler, hyperbola
•bachelor
•paddler, straddler
•mandala • panhandler • meddler
•ladler • wheedler
•diddler, piddler, riddler, tiddler, twiddler
•coddler, modeller (US modeler), toddler, twaddler, waddler
•fondler, gondola
•yodeller (US yodeler)
•doodler
•muddler, puddler
•hurdler • waffler
•shuffler, snuffler
•haggler, straggler
•mangler, wangler
•finagler
•giggler, wiggler, wriggler
•smuggler, struggler
•pergola • heckler
•Agricola, Nicola, pickler, tickler, tricolour (US tricolor)
•chronicler
•snorkeller (US snorkeler)
•chuckler
•enameller (US enameler)
•signaller (US signaler)
•tunneller (US tunneler)
•grappler • stapler
•stippler, tippler
•Coppola
•gospeller (US gospeler)
•cupola
•caroller (US caroler)
•Kerala
•quarreller (US quarreler)
•chancellor
•penciller (US penciler)
•whistler
•battler, prattler, rattler, tattler
•dismantler • startler
•fettler, settler, settlor
•belittler, victualler (US victualer)
•hospitaller (US hospitaler)
•bottler, throttler
•hosteller (US hosteler)
•caviller (US caviler), traveller (US traveler)
•marveller (US marveler)
•leveller (US leveler), reveller (US reveler)
•driveller (US driveler), sniveller (US sniveler)
•groveller (US groveler)
•shoveler, shoveller
•chiseller (US chiseler), sizzler
•bamboozler, methuselah
•guzzler
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Hyperbola
Hyperbola
A hyperbola is a curve formed by the intersection of a right circular cone and a plane (see Figure 1). When the plane cuts both nappes of the cone, the intersection is a hyperbola. Because the plane is cutting two nappes, the curve it forms has two Ushaped branches opening in opposite directions.
Other definitions
A hyperbola can be defined in several other ways, all of them mathematically equivalent:
 A hyperbola is a set of points P such that PF_{1}PF_{2} = ±C, where C is a constant and F_{1} and F_{2} are fixed points called the “foci” (see Figure 2). That is, a hyperbola is the set of points the difference of whose distances from two fixed points is constant. The positive value of ±C gives one branch of the hyperbola; the negative value, the other branch.
 A hyperbola is a set of points whose distances from a fixed point (the “focus”) and a fixed line (the “directrix”) are in a constant ratio (the “eccentricity”). That is, PF/PD = e (see Figure 3). For this set of points to be a hyperbola, e has to be greater than 1. This definition gives only one branch of the hyperbola.
 A hyperbola is a set of points (x,y) on a Cartesian coordinate plane satisfying an equation of the form x^{2}/A^{2}y^{2}/B^{2} = ±1. The equation xy = k also represents a hyperbola, but of eccentricity not equal to 2. Other seconddegree equations can represent hyperbolas, but these two forms are the simplest. When the positive value in ± 1 is used, the hyperbola opens to the left and right. When the negative value is used, the hyperbola opens up and down.
Features
When a hyperbola is drawn as in Figure 4, the line through the foci, F_{1} and F_{2}, is the “transverse axis.” V_{1} and V_{2} are the “vertices,” and C the “center.” The transverse axis also refers to the distance, V_{1}V_{2}, between the vertices.
The ratio CF_{1}/CV_{1} (or CF_{2}/CV_{2}) is the “eccentricity” and is numerically equal to the eccentricity e in the focusdirectrix definition.
The lines CR and CQ are asymptotes. An asymptote is a straight line which the hyperbola approaches more and more closely as it extends farther and farther from the center. The point Q has been located so that it is the vertex of a right triangle, one of whose legs is CV_{2}, and whose hypotenuse CQ equals CF_{2}. The point R is similarly located.
The line ST, perpendicular to the transverse axis at C, is called the “conjugate axis.” The conjugate axis also refers to the distance ST, where SC = CT = QV_{2}.
A hyperbola is symmetric about both its transverse and its conjugate axes.
When a hyperbola is represented by the equation x^{2}/A^{2}y^{2}/B^{2} = 1, the xaxis is the transverse axis and the yaxis is the conjugate axis. These axes, when thought of as distances rather than lines, have lengths 2A and 2B respectively. The foci are at
The equations of the asymptotes are y = Bx/A and y = Bx/A. (Notice that the constant 1 in the equation above is positive. If it were 1, the yaxis would be the transverse axis and the other points would change accordingly. The asymptotes would be the same, however. In fact, the hyperbolas x^{2}/A^{2}  y^{2}/B^{2} = 1 and x^{2}/A^{2}  y^{2}/B^{2} = 1 are called “conjugate hyperbolas.”) Hyperbolas whose asymptotes are perpendicular to each other are called “rectangular” hyperbolas. The hyperbolas xy = k and x^{2}  y^{2} = ± C^{2} are rectangular hyperbolas. Their eccentricity is ≠2. Such hyperbolas are geometrically similar, as are all hyperbolas of a given eccentricity.
If one draws the angle F_{1}PF_{2} the tangent to the hyperbola at point P will bisect that angle.
Drawing hyperbolas
Hyperbolas can be sketched quite accurately by first locating the vertices, the foci, and the asymptotes.
Starting with the axes, locate the vertices and foci. Draw a circle with its center at C, passing through the two foci. Draw lines through the vertices perpendicular to the transverse axis. This determines four points, which are corners of a rectangle. These diagonals are the asymptotes.
Using the vertices and asymptotes as guides, sketch in the hyperbola as shown in Figure 5. The hyperbola approaches the asymptotes, but never quite reaches them. Its curvature, therefore, approaches, but never quite reaches, that of a straight line.
If the lengths of the transverse and conjugate axes are known, the rectangle in Figure 5 can be drawn without using the foci, since the rectangle’s length and width are equal to these axes.
One can also draw hyperbolas by plotting points on a coordinate plane. In doing this, it helps to draw the asymptotes, whose equations are given above.
Uses
Hyperbolas have many uses, both mathematical and practical. The hyperbola y = 1/x is sometimes used in the definition of the natural logarithm. In Figure 6 the logarithm of a number n is represented by the shaded area, that is, by the area bounded by the xaxis, the line x = 1, the line x = n, and the hyperbola. Of course one needs calculus to compute this area, but there are techniques for doing so.
The coordinates of the point (x,y) on the hyperbola x^{2}  y^{2} = 1 represent the hyperbolic cosine and hyperbolic sine functions. These functions bear the same relationship to this particular hyperbola that the ordinary cosine and sine functions bear to a unit circle:
x = cosh u = (e^{U} + e^{u})2
y = sinh u = (e^{U}  e^{u})2
Unlike ordinary sines and cosines, the values of the hyperbolic functions can be represented with simple exponential functions, as shown above. That these representations work can be checked by substituting them in the equation of the hyperbola. The parameter u is also related to the hyperbolas. It is twice the shaded area in Figure 7.
The definition PF1  PF_{2} = ± C, of a hyperbola is used directly in the LORAN navigational system. A ship at P receives simultaneous pulsed radio signals from stations at A and B. It cannot measure the time it takes for the signals to arrive from each of these stations, but it can measure how much longer it takes for the signal to arrive from one station than from the
other. It can therefore compute the difference PA  PB in the distances. This locates the ship somewhere along a hyperbola with foci at A and B, specifically the hyperbola with that constant difference. In the same way, by timing the difference in the time it takes to receive simultaneous signals from stations B and C, it can measure the difference in the distances PB and PC. This puts it somewhere on a second hyperbola with B and C as foci and PC  PB as the constant difference. The ship’s position is where these two hyperbolas cross (Figure 8). Maps with grids of crossing hyperbolas are available to the ship’s navigator for use in areas served by these stations.
Resources
BOOKS
Gullberg, Jan, and Peter Hilton. Mathematics: From the Birth of Numbers. W.W. Norton & Company, 1997.
KEY TERMS
Foci —Two fixed points on the transverse axis of a hyperbola. Any point on the hyperbola is always a fixed amount farther from one focus than from the other.
Hyperbola —A conic section of two branches, satisfying one of several definitions.
Vertices —The two points where the hyperbola crosses the transverse axis.
Hahn, Liangshin. Complex Numbers and Geometry. 2nd ed. The Mathematical Association of America, 1996.
Hilbert, D. and CohnVossen, S. Geometry and the Imagination. New York: Chelsea Publishing Co. 1952.
Larson, Ron. Calculus With Analytic Geometry. Boston: Houghton Mifflin College, 2002.
J. Paul Moulton
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Hyperbola
Hyperbola
A hyperbola is a curve formed by the intersection of a right circular cone and a plane (see Figure 1). When the plane cuts both nappes of the cone, the intersection is a hyperbola. Because the plane is cutting two nappes, the curve it forms has two Ushaped branches opening in opposite directions.
Other definitions
A hyperbola can be defined in several other ways, all of them mathematically equivalent:
 A hyperbola is a set of points P such that PF1 PF2 = ± C, where C is a constant and F1 and F2 are fixed points called the "foci" (see Figure 2). That is, a hyperbola is the set of points the difference of whose distances from two fixed points is constant. The positive value of ± C gives one branch of the hyperbola; the negative value, the other branch.
 A hyperbola is a set of points whose distances from a fixed point (the "focus") and a fixed line (the "directrix") are in a constant ratio (the "eccentricity"). That is, PF/PD = e (see Figure 3). For this set of points to be a hyperbola, e has to be greater than 1. This definition gives only one branch of the hyperbola.
 A hyperbola is a set of points (x,y) on a Cartesian coordinate plane satisfying an equation of the form x2/A2  y2/B2 = ± 1. The equation xy = k also represents a hyperbola, but of eccentricity not equal to 2. Other seconddegree equations can represent hyperbolas, but these two forms are the simplest. When the positive value in ± 1 is used, the hyperbola opens to the left and right. When the negative value is used, the hyperbola opens up and down.
Features
When a hyperbola is drawn as in Figure 4, the line through the foci, F1 and F2, is the "transverse axis." V1 and V2 are the "vertices," and C the "center." The transverse axis also refers to the distance ,V1V2, between the vertices.
The ratio CF1/CV1 (or CF2/CV2) is the "eccentricity" and is numerically equal to the eccentricity e in the focusdirectrix definition.
The lines CR and CQ are asymptotes. An asymptote is a straight line which the hyperbola approaches more and more closely as it extends farther and farther from the center. The point Q has been located so that it is the vertex of a right triangle, one of whose legs is CV2, and whose hypotenuse CQ equals CF2. The point R is similarly located.
The line ST, perpendicular to the transverse axis at C, is called the "conjugate axis." The conjugate axis also refers to the distance ST, where SC = CT = QV2.
A hyperbola is symmetric about both its transverse and its conjugate axes.
When a hyperbola is represented by the equation x2/A2  y2/B2 = 1, the xaxis is the transverse axis and the yaxis is the conjugate axis. These axes, when thought of as distances rather than lines, have lengths 2A and 2B respectively. The foci are at
The equations of the asymptotes are y = Bx/A and y = Bx/A. (Notice that the constant 1 in the equation above is positive. If it were 1, the yaxis would be the transverse axis and the other points would change accordingly. The asymptotes would be the same, however. In fact, the hyperbolas x2/A2  y2/B2 = 1 and x2/A2  y2/B2 = 1 are called "conjugate hyperbolas.") Hyperbolas whose asymptotes are perpendicular to each other are called "rectangular" hyperbolas. The hyperbolas xy = k and x2  y2 = ± C2 are rectangular hyperbolas. Their eccentricity is ≠ 2. Such hyperbolas are geometrically similar, as are all hyperbolas of a given eccentricity.
If one draws the angle F1PF2 the tangent to the hyperbola at point P will bisect that angle. ——
Drawing hyperbolas
Hyperbolas can be sketched quite accurately by first locating the vertices, the foci, and the asymptotes. Starting with the axes, locate the vertices and foci. Draw a circle with its center at C, passing through the two foci. Draw lines through the vertices perpendicular to the transverse axis. This determines four points, which are corners of a rectangle . These diagonals are the asymptotes.
Using the vertices and asymptotes as guides, sketch in the hyperbola as shown in Figure 5. The hyperbola approaches the asymptotes, but never quite reaches them. Its curvature, therefore, approaches, but never quite reaches, that of a straight line.
If the lengths of the transverse and conjugate axes are known, the rectangle in Figure 5 can be drawn without
using the foci, since the rectangle's length and width are equal to these axes.
One can also draw hyperbolas by plotting points on a coordinate plane. In doing this, it helps to draw the asymptotes, whose equations are given above.
Uses
Hyperbolas have many uses, both mathematical and practical. The hyperbola y = 1/x is sometimes used in the definition of the natural logarithm. In Figure 6 the logarithm of a number n is represented by the shaded area, that is, by the area bounded by the xaxis, the line x = 1, the line x = n, and the hyperbola. Of course one needs calculus to compute this area, but there are techniques for doing so.
The coordinates of the point (x,y) on the hyperbola x2 y2 = 1 represent the hyperbolic cosine and hyperbolic sine functions. These functions bear the same relationship to this particular hyperbola that the ordinary cosine and sine functions bear to a unit circle:
Unlike ordinary sines and cosines, the values of the hyperbolic functions can be represented with simple exponential functions, as shown above. That these representations work can be checked by substituting them in the equation of the hyperbola. The parameter u is also related to the hyperbolas. It is twice the shaded area in Figure 7.
The definition PF1  PF2 = ± C, of a hyperbola is used directly in the LORAN navigational system. A ship at P receives simultaneous pulsed radio signals from stations at A and B. It cannot measure the time it takes for the signals to arrive from each of these stations, but it can measure how much longer it takes for the signal to arrive from one station than from the other. It can therefore compute the difference PA  PB in the distances. This locates the ship somewhere along a hyperbola with foci at A and B, specifically the hyperbola with that constant difference. In the same way, by timing the difference in the time it takes to receive simultaneous signals from stations B and C, it can measure the difference in the distances PB and PC. This puts it somewhere on a second hyperbola with B and C as foci and PC  PB as the constant difference. The ship's position is where these two hyperbolas cross (Figure 8). Maps with grids of crossing hyperbolas are available to the ship's navigator for use in areas served by these stations.
Resources
books
Gullberg, Jan, and Peter Hilton. Mathematics: From the Birth of Numbers. W.W. Norton & Company, 1997.
Hahn, Liangshin. Complex Numbers and Geometry. 2nd ed. The Mathematical Association of America, 1996.
Hilbert, D., and S. CohnVossen. Geometry and the Imagination. New York: Chelsea Publishing Co. 1952.
Larson, Ron. Calculus With Analytic Geometry. Boston: Houghton Mifflin College, 2002.
J. Paul Moulton
KEY TERMS
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Foci
—Two fixed points on the transverse axis of a hyperbola. Any point on the hyperbola is always a fixed amount farther from one focus than from the other.
 Hyperbola
—A conic section of two branches, satisfying one of several definitions.
 Vertices
—The two points where the hyperbola crosses the transverse axis.
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Citation styles
Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).
Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.
Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the mostrecent information available at these sites:
Modern Language Association
The Chicago Manual of Style
http://www.chicagomanualofstyle.org/tools_citationguide.html
American Psychological Association
Notes:
 Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
 In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.