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Hyperbolas

Definitions
General Equation of a Hyperbola
Standard Forms of a Hyperbola
Reduction to Standard Form
Hyperbola from Vertices and Eccentricity
Hyperbola from Foci and Eccentricity
Hyperbola from Focus and Directrix
Parametric Equations
Explorations

The equations of a hyperbola are in many ways similar to those of an ellipse, the forms often only differing by a + or - sign. The properties and characteristics of a hyperbola, however, are somewhat less intuitive than an ellipse, possibly because the curve has two disjoint branches or because it extends to infinity. This chapter describes the detailed mathematics of a hyperbola.

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<<Descarta2D`

Definitions [Top]

A hyperbola is the locus of a point that moves so that the ratio of its distance from a fixed point and from a fixed line is a constant greater than one. As with the parabola and ellipse, a focus, directrix and eccentricity are associated with the curve as shown in Table [hyp:tbl01].

"hyp_1.gif"

Hyperbola definition.

"hyp_2.gif"

Hyperbola definition.

Consider the line through the focus perpendicular to the directrix. From the definition PF/PD=e/1 there are obviously two points V and "hyp_3.gif" which divide the (undirected) segment FD, internally and externally respectively, in the ratio of e/1. Therefore, V and "hyp_4.gif" are points (on opposite sides of D) on the hyperbola; they are called the vertices. The segment "hyp_5.gif" is called the transverse axis. By symmetry, there is another point "hyp_6.gif" and another line "hyp_7.gif" such that "hyp_8.gif" and "hyp_9.gif" would serve in the definition of this curve. Thus, a hyperbola has two foci and two directrices associated in pairs F, D and "hyp_10.gif", "hyp_11.gif". The midpoint of "hyp_12.gif", which is also the midpoint of "hyp_13.gif", is called the center C. There are two tangent lines through C whose points of contact are at an infinite distance from C. These are called the asymptotes of the hyperbola. The focal chord perpendicular to the transverse axis is called the latus rectum.

A line through C perpendicular to the transverse axis does not intersect the hyperbola in real points. But the portion of it, bisected by C, which is equal in length to the parallel segment through V contained between the asymptotes is called the conjugate axis.

Example: Plot the hyperbola with center at coordinates (2,1), transverse axis length of 1, conjugate axis length of 3/4 and rotated "hyp_14.gif" (π/6 radians) about the center point.

Solution: Hyperbola2D[{h,k},a,b,θ] is the standard representation of a hyperbola in Descarta2D. The hyperbola is centered at coordinates (h,k), has semi-transverse axis of a, semi-conjugate axis of b and is rotated about the center point by an angle θ (the semi-transverse axis is half the length of the transverse axis; the semi-conjugate axis is half the length of the conjugate axis).

Sketch2D[{Hyperbola2D[{2,1},1,3/4,Pi/6]}]

"hyp_15.gif"

General Equation of a Hyperbola [Top]

Take any point "hyp_16.gif" as focus and any line, "hyp_17.gif" as directrix, where "hyp_18.gif". The normalized form of the line is used to simplify the derivation. By definition the equation of the hyperbola is

"hyp_19.gif"

which may be expanded to

"hyp_20.gif"
"hyp_21.gif"

This is of the form "hyp_22.gif", an equation of the second degree. Moreover, it can be verified that "hyp_23.gif" (when e>1).

Therefore, a necessary condition that "hyp_24.gif" represent a hyperbola is that "hyp_25.gif". The general equation reveals that if the defining directrix line is parallel to one of the coordinate axes then B=0, since either "hyp_26.gif" or "hyp_27.gif" will be zero. The equation of a hyperbola in this position will have no xy term.

Standard Forms of a Hyperbola [Top]

By an appropriate choice of coordinate axes the general equation of a hyperbola can be reduced to one of the following standard forms.

"hyp_28.gif"

Hyperbola in standard position (x-axis).

"hyp_29.gif"

Hyperbola in standard position (y-axis).

Transverse Axis Parallel to the x-Axis

The equation of a hyperbola in standard position whose transverse axis is parallel to the x-axis and whose center is at the origin is

"hyp_30.gif"

where a and b are the lengths of the semi-transverse and semi-conjugate axes, respectively. If the hyperbola is centered at (h,k), then the equation is

"hyp_31.gif"

as shown in Figure [hyp:fig3]. When a hyperbola is in this special position, the formulas for the important points, lines and constants associated with the hyperbola are simply determined and are summarized in Table [hyp:tbl02].

The lengths of the transverse axis, conjugate axis, focal chord and the value of the eccentricity are independent of the origin and are also given in Table [hyp:tbl02]. Note that the equations of the asymptotes can be obtained directly from the equation of the hyperbola in standard form by replacing the one on the right-hand side of the equation with a zero. The left-hand side of the equation will then factor into two linear terms which are the asymptotes of the hyperbola.

"hyp_32.gif"

Hyperbola definition (x- and y-axis).

Transverse Axis Parallel to the y-Axis

The equation of a hyperbola in standard position whose transverse axis is parallel to the y-axis and whose center is at the origin is

"hyp_33.gif"

where a and b are the lengths of the semi-transverse and semi-conjugate axes, respectively. If the hyperbola is centered at (h,k), then the equation is

"hyp_34.gif"

as shown in Figure [hyp:fig4]. When a hyperbola is in this special position, the formulas for the important points, lines and constants associated with the hyperbola are simply determined and are summarized in Table [hyp:tbl02]

The lengths of the semi-transverse axis, semi-conjugate axis, focal chord and the value of the eccentricity are independent of the origin and are also shown in Table [hyp:tbl02]. These constants have the same values as a hyperbola whose transverse axis is parallel to the x-axis.

Conjugate and Rectangular Hyperbolas

Two hyperbolas with the same center are conjugate hyperbolas if the transverse axis of one coincides with the conjugate axis of the other. The equations of two conjugate hyperbolas H and "hyp_35.gif" in standard form are shown in Table [hyp:tbl07]. It is evident that if a is the semi-transverse axis of H, then a is the semi-conjugate axis of "hyp_36.gif", and vice versa. Conjugate hyperbolas have the same asymptotes and their foci lie on a circle with center at the center of the hyperbolas.

"hyp_37.gif"

Conjugate hyperbolas.

Example: Write the equation of the hyperbola whose center is (-2,1), transverse axis length 6 (parallel to the x-axis), and conjugate axis length 8. Determine its eccentricity, foci and vertices. Find the equations of its directrices and asymptotes. Plot the geometric objects.

Solution: The equation can be written directly using the standard form as

"hyp_38.gif"

In Descarta2D this hyperbola is written as Hyperbola2D[{-2,1},3,4,0].

h1=Hyperbola2D[{-2,1},3,4,0];

The Descarta2D function Eccentricity2D[hyperbola] returns the eccentricity of the hyperbola.

Eccentricity2D[h1]

"hyp_39.gif"

The Descarta2D function Foci2D[hyperbola] returns a list of the two focus points; the function Vertices2D[hyperbola] returns a list of the two vertex points; the function Directrices2D[hyperbola] returns a list of the two directrix lines; the function Asymptotes2D[hyperbola] returns a list of the two asymptote lines.

objs=Map[( #[h1] )&,
     {Foci2D,Vertices2D,Directrices2D,Asymptotes2D}]

"hyp_40.gif"

Sketch2D[{h1,objs},
          CurveLength2D->40,
         PlotRange->{{-14,10},{-9,11}}]

"hyp_41.gif"

Example: Plot the hyperbola whose equation is "hyp_42.gif" along with its conjugate.

Solution: The function Loci2D[quad] constructs a list containing the objects represented by a quadratic equation; Hyperbola2D[hyperbola,Conjugate2D] constructs the conjugate of a hyperbola.

{h1}=Loci2D[Quadratic2D[4,0,-1,0,0,36]]

"hyp_43.gif"

h2=Hyperbola2D[h1,Conjugate2D]

"hyp_44.gif"

f={{f1a,f1b}=Foci2D[h1],{f2a,f2b}=Foci2D[h2]};
c1=Circle2D[f1a,f1b,f2a];IsOn2D[f2b,c1]

"hyp_45.gif"

The statement IsOn2D[f2b,c1], by returning True, shows that the foci of both hyperbolas are on a common circle.

Sketch2D[{h1,h2,f,c1},
          CurveLength2D->40,
         PlotRange->{{-12,12},{-10,10}}]

"hyp_46.gif"

A rectangular (or equilateral) hyperbola is one in which the transverse and conjugate axes are equal in length, in which case the asymptotes are at right angles to each other.

Reduction to Standard Form [Top]

The most general equation of a hyperbola with no xy term (and hence one whose axes are parallel to the coordinate axes) is of the form

"hyp_47.gif"

The condition "hyp_48.gif" reduces to AC<0 which implies that A and C are of opposite sign. This equation can be reduced to one of the standard forms by completing the square.

Example: Reduce "hyp_49.gif" to standard form and plot.

Solution: The Descarta2D function Loci2D[quad] constructs a list containing the objects represented by the quadratic.

h1=Loci2D[Quadratic2D[1,0,-1,-2,-1,1]]

"hyp_50.gif"

The equation in standard form is

"hyp_51.gif"

This is a rectangular hyperbola with "hyp_52.gif".

Sketch2D[{h1}]

"hyp_53.gif"

Hyperbola from Vertices and Eccentricity [Top]

Suppose we are given the two vertices, "hyp_54.gif" and "hyp_55.gif" and the eccentricity, e, of a hyperbola and we wish to find the standard equation of the hyperbola. The center point (h,k) of the hyperbola is clearly the midpoint between the vertices and is given by

"hyp_56.gif"

The length of the semi-transverse axis, a, is one-half the distance between the vertices, yielding "hyp_57.gif". The eccentricity is given by

"hyp_58.gif"

so, solving for b gives the length of the semi-conjugate axis as

"hyp_59.gif"

The line through the two vertex points determines the rotation angle of the hyperbola as

"hyp_60.gif"

Example: Find the hyperbola whose vertices are (4,2) and (-2,1), and whose eccentricity is 3/2.

Solution: The Descarta2D function Hyperbola2D[{point,point},e] returns the hyperbola whose vertices are the given points with the specified eccentricity.

p1=Point2D[{4,2}];
p2=Point2D[{-2,1}];
h1=Hyperbola2D[{p1,p2},3/2] //N

"hyp_61.gif"

Sketch2D[{p1,p2,h1}]

"hyp_62.gif"

Hyperbola from Foci and Eccentricity [Top]

It is evident from Table [hyp:tbl02] that the distance between the foci of a hyperbola is given by "hyp_63.gif" and that the distance between the vertices is "hyp_64.gif". Therefore, the eccentricity, e, given by

"hyp_65.gif"

is the ratio of the distance between the foci to the distance between the vertices. This relationship allows us to construct a hyperbola by specifying the two foci and the eccentricity. The semi-transverse axis length, a, is given by "hyp_66.gif" and the semi-conjugate axis length is "hyp_67.gif". The center point of the hyperbola is clearly the midpoint of the two foci and the angle of rotation is "hyp_68.gif", where "hyp_69.gif" and "hyp_70.gif" are the coordinates of the foci.

Example: Find the hyperbola whose foci are (-1,-1) and (1,1) and whose eccentricity is 3/2.

Solution: The function Hyperbola2D[point,point,e] constructs a hyperbola given the two foci points and the eccentricity.

h1=Hyperbola2D[Point2D[{-1,-1}],Point2D[{1,1}],3/2]

"hyp_71.gif"

{Foci2D[h1],Eccentricity2D[h1]}

"hyp_72.gif"

Hyperbola from Focus and Directrix [Top]

Given the focus point "hyp_73.gif", the directrix line L≡px+qy+r=0 and the eccentricity, e>1, of a hyperbola we wish to determine the standard equation of the hyperbola. The rotation angle of the hyperbola is the angle the line perpendicular to L makes with the +x-axis and is given by "hyp_74.gif". The distance, d, from F to L is given by

"hyp_75.gif"

It is clear from Table [hyp:tbl02] that the distance from F to L is also given by d=ae-a/e. Solving for a (the length of the semi-transverse axis) yields

"hyp_76.gif"

Table [hyp:tbl02] shows that the eccentricity, e, is related to the lengths of the semi-transverse and semi-conjugate axes, a and b, respectively, by

"hyp_77.gif"

Solving this equation for b yields

"hyp_78.gif"

Table [hyp:tbl02] reveals that the distance from the focus F to the center C(h,k) is given by ae. If F' is the projection of F onto L, then we can find the center point C(h,k) of the hyperbola by offsetting F in the direction from F to F' a distance ae. This computation is easily accomplished using Descarta2D and is provided in the exploration hypfd.html. The defining constants of the hyperbola so computed are

"hyp_79.gif"

"hyp_80.gif"

where

"hyp_81.gif"

Example: Find the hyperbola whose focus is (3,2), directrix line is x-y+2=0 and eccentricity is 5.

Solution: The Descarta2D function Hyperbola2D[point,line,e] constructs a hyperbola from the focus, directrix and eccentricity.

h1=Hyperbola2D[p1=Point2D[{3,2}],l2=Line2D[1,-1,2],5]

"hyp_82.gif"

{Foci2D[h1],
Directrices2D[h1],
Eccentricity2D[h1]} //Simplify

"hyp_83.gif"

Parametric Equations [Top]

The standard form of a hyperbola used in Descarta2D has the equation

"hyp_84.gif"

where (h,k) is the center of the hyperbola, and a and b are the lengths of the semi-transverse and semi-conjugate axes, respectively. The axis of this hyperbola is parallel to the x-axis and the hyperbola opens to the right and left. Hyperbolas in other orientations are obtained by applying a rotation, θ, to the standard hyperbola. The parametric equations for a hyperbola are similar to those of an ellipse, except hyperbolic functions are used instead of standard trigonometric functions. The parametric equations are

x=h+cosht   and   y=k+sinht.

The parameter value t=0 produces the vertex point on the right branch of the hyperbola. Increasing values of t produce points above and to the right of this vertex. Negative values of t produce points that correspond to positive t values reflected in the transverse axis of the hyperbola. All of the points on the right branch need to be reflected in the conjugate axis of the hyperbola to produce the left branch of the curve.

In Descarta2D the parametric equations of a hyperbola are scaled by a factor s so that the end points of the focal chord are at the parameter values -1 and +1. Specifically, the equations used in Descarta2D are

x=h+acosh(st)   and   y=k+bsinh(st)

where

"hyp_85.gif"

and e is the eccentricity of the hyperbola. The validity of these equations can be verified by direct substitution.

Example: Plot eight points at equal parameter values on the upper and lower portions of the right branch of the hyperbola "hyp_86.gif".

Solution: The command Hyperbola2D[{h,k},a,b,θ][t] returns the coordinates at parameter t on the hyperbola.

h1=Hyperbola2D[{0,0},2,Sqrt[2],0];
pts1=Map[Point2D[h1[#/3]]&,Range[0,7]];
pts2=Map[Point2D[h1[#/3]]&,Range[-7,0]];
pr=PlotRange->{{-6,10},{-5,5}};
{Sketch2D[{h1,pts1},pr],Sketch2D[{h1,pts2},pr]}

"hyp_87.gif"

As with the ellipse, a pair of rational equations may be used as the parametric equations for a hyperbola. The hyperbola

"hyp_88.gif"

has the parametric equations

"hyp_89.gif"

Values of t in the range 0≤t<1 generate coordinates on the hyperbola in the first quadrant. The other portions of the curve can be generated by reflecting the coordinates generated by these equations.

Example: Plot the hyperbola "hyp_90.gif" using the rational parametric equations in the parameter range -1/2≤t≤1/2.

Solution: The Mathematica function ParametricPlot plots curves defined by parametric equations.

Clear[t];
ParametricPlot[{5*(1+t^2)/(1-t^2),2*1*t/(1-t^2)},
               {t,-1/2,1/2},AspectRatio->Automatic]

"hyp_91.gif"

Explorations [Top]

Length of Hyperbola Focal Chord. [hyplen.html]

Prove that the length of the focal chord of a hyperbola is "hyp_92.gif", where a is the length of the semi-transverse axis and b is the length of the semi-conjugate axis.

Focal Distances of a Hyperbola. [hyp2a.html]

Show that the difference of the distances from the two foci to any point on a hyperbola is 2a, where a is the length of the semi-transverse axis.

Hyperbola from Focus and Directrix. [hypfd.html]

Show that the hyperbola with focus "hyp_93.gif", directrix L≡px+qy+r=0 and eccentricity, e>1 is defined by the constants

"hyp_94.gif"

"hyp_95.gif"

where

"hyp_96.gif"

Rectangular Hyperbola Distances. [hypinv.html]

Show that the distance of any point on a rectangular hyperbola from its center varies inversely as the perpendicular distance from its polar to the center.

Eccentricities of Conjugate Hyperbolas. [hypeccen.html]

Show that if "hyp_97.gif" and "hyp_98.gif" are the eccentricities of a hyperbola and its conjugate, then "hyp_99.gif".

Polar Equation of a Hyperbola [polarhyp.html]

Show that the polar equation of a hyperbola with a horizontal transverse axis and centered at (0,0) is given by

"hyp_100.gif"

where a and b are the lengths of the semi-transverse and semi-conjugate axes, respectively.

Trigonometric Parametric Equations [hyptrig.html]

Show that the parametric equations

x=a  secθ   and   y=b  tanθ

represent the hyperbola

"hyp_101.gif"


Copyright © 1999-2007 Donald L. Vossler, Descarta2D Publishing
www.Descarta2D.com