# 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 U-shaped 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 second-degree 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 focus-directrix 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 x-axis is the transverse axis and the y-axis 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 y-axis 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 x-axis, 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, Liang-shin. *Complex Numbers and Geometry*. 2nd ed. The Mathematical Association of America, 1996.

Hilbert, D. and Cohn-Vossen, 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

# 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 U-shaped 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 second-degree 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 focus-directrix 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 x-axis is the transverse axis and the y-axis 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 y-axis 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 x-axis, 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, Liang-shin. *Complex Numbers and Geometry.* 2nd ed. The Mathematical Association of America, 1996.

Hilbert, D., and S. Cohn-Vossen. *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.

# 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.

# 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.

# 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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