11-3 Hyperbola
Leading edge
Fuselage
799
and 1.00 foot in front of it. The chord is 1.00 foot shorter
than the major axis.
Trailing edge
Elliptical
wings and tail
(A) If the straight-line leading edge is parallel to the major
axis of the ellipse and is 1.14 feet in front of it, and if
the leading edge is 46.0 feet long (including the width
of the fuselage), find the equation of the ellipse. Let the
x axis lie along the major axis (positive right), and let
the y axis lie along the minor axis (positive forward).
(B) How wide is the wing in the center of the fuselage (assuming the wing passes through the fuselage)?
Compute quantities to 3 significant digits.
★
42. Naval Architecture. Currently, many high-performance
racing sailboats use elliptical keels, rudders, and main sails
for the same reasons stated in Problem 41—less drag along
the trailing edge. In the accompanying figure, the ellipse
containing the keel has a 12.0-foot major axis. The straightline leading edge is parallel to the major axis of the ellipse
SECTION
11-3
Rudder
Keel
(A) Find the equation of the ellipse. Let the y axis lie along
the minor axis of the ellipse, and let the x axis lie along
the major axis, both with positive direction upward.
(B) What is the width of the keel, measured perpendicular
to the major axis, 1 foot up the major axis from the bottom end of the keel?
Compute quantities to 3 significant digits.
Hyperbola
•
•
•
•
Definition of a Hyperbola
Drawing a Hyperbola
Standard Equations and Their Graphs
Applications
As before, we start with a coordinate-free definition of a hyperbola. Using this definition, we show how a hyperbola can be drawn and we derive standard equations for
hyperbolas specially located in a rectangular coordinate system.
• Definition of
a Hyperbola
The following is a coordinate-free definition of a hyperbola:
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11
Additional Topics in Analytic Geometry
DEFINITION 1
Hyperbola
A hyperbola is the set of all points P in a
plane such that the absolute value of the difference of the distances of P to two fixed
points in the plane is a positive constant.
Each of the fixed points, F and F, is called
a focus. The intersection points V and V of
the line through the foci and the two
branches of the hyperbola are called vertices, and each is called a vertex. The line
segment VV is called the transverse axis.
The midpoint of the transverse axis is the
center of the hyperbola.
• Drawing a
Hyperbola
d1 d2 Constant
P
d2
d1
F
V
V
F
Thumbtacks, a straightedge, string, and a pencil are all that are needed to draw a
hyperbola (see Fig. 1). Place two thumbtacks in a piece of cardboard—these form the
foci of the hyperbola. Rest one corner of the straightedge at the focus F so that it is
free to rotate about this point. Cut a piece of string shorter than the length of the
straightedge, and fasten one end to the straightedge corner A and the other end to the
thumbtack at F. Now push the string with a pencil up against the straightedge at B.
Keeping the string taut, rotate the straightedge about F, keeping the corner at F. The
resulting curve will be part of a hyperbola. Other parts of the hyperbola can be drawn
by changing the position of the straightedge and string. To see that the resulting curve
meets the conditions of the definition, note that the difference of the distances BF
and BF is
BF
BF BF
BA BF BA
AF
(BF BA)
String
Straightedge
length length Constant
FIGURE 1 Drawing a hyperbola.
A
B
F
F
String
11-3 Hyperbola
• Standard Equations
and Their Graphs
y
d1
F
(c, 0)
P (x, y)
d2
x
F (c, 0)
Using the definition of a hyperbola and the distance-between-two-points formula, we
can derive the standard equations for a hyperbola located in a rectangular coordinate
system. We start by placing a hyperbola in the coordinate system with the foci on the
x axis equidistant from the origin at F(c, 0) and F(c, 0), c 0, as in Figure 2.
Just as for the ellipse, it is convenient to represent the constant difference by 2a,
a 0. Also, the geometric fact that the difference of two sides of a triangle is always
less than the third side can be applied to Figure 2 to derive the following useful result:
d
1
d 2 2c
2a 2c
ac
c0
d1 d2 Positive constant
FIGURE 2 Hyperbola with foci on
the x axis.
801
(1)
We will use this result in the derivation of the equation of a hyperbola, which we now
begin.
Referring to Figure 2, the point P(x, y) is on the hyperbola if and only if
d
d(P, F
) d(P, F) 2a
y (x c) y 2a
1
(x c)
2
2
d 2 2a
2
2
After eliminating radicals and absolute value signs by appropriate use of squaring and
simplifying, another good exercise for you, we have
(c2 a2)x2 a2y2 a2(c2 a2)
2
(2)
2
x
y
2
1
2
a
c a2
(3)
Dividing both sides of equation (2) by a2(c2 a2) is permitted, since neither a2 nor
c2 a2 is 0. From equation (1), a c; thus, a2 c2 and c2 a2 0. The constant
a was chosen positive at the beginning.
To simplify equation (3) further, we let
b2 c2 a2
b0
(4)
to obtain
x2 y2
1
a2 b2
(5)
From equation (5) we see that the x intercepts, which are also the vertices, are
x a and there are no y intercepts. To see why there are no y intercepts, let
x 0 and solve for y:
02 y2
1
a2 b2
y2 b2
y b2
An imaginary number
802
11
Additional Topics in Analytic Geometry
y
d1
If we start with the foci on the y axis at F(0, c) and F(0, c) as in Figure 3,
instead of on the x axis as in Figure 2, then, following arguments similar to those
used for the first derivation, we obtain
P(x, y)
F(0, c)
d2
y2 x2
1
a2 b2
x
(6)
where the relationship among a, b, and c remains the same as before:
F
(0, c)
b2 c2 a2
c0
d1 d2 Positive constant
FIGURE 3 Hyperbola with foci on
the y axis.
(7)
The center is still at the origin, but the transverse axis is now on the y axis.
As an aid to graphing equation (5), we solve the equation for y in terms of x,
another good exercise for you, to obtain
b
y x
a
1
a2
x2
(8)
As x changes so that x becomes larger, the expression 1 (a2/x2) within the radical approaches 1. Hence, for large values of x , equation (5) behaves very much like
the lines
b
y x
a
(9)
These lines are asymptotes for the graph of equation (5). The hyperbola approaches
these lines as a point P(x, y) on the hyperbola moves away from the origin (see Fig.
4). An easy way to draw the asymptotes is to first draw the rectangle as in Figure 4,
then extend the diagonals. We refer to this rectangle as the asymptote rectangle.
FIGURE 4 Asymptotes.
Asymptote
b
y x
a
Asymptote
b
y x
a
y
b
a
0
a
y2
x2
1
a2 b2
x
b
Starting with equation (6) and proceeding as we did for equation (5), we obtain
the asymptotes for the graph of equation (6):
a
y x
b
(10)
The perpendicular bisector of the transverse axis, extending from one side of the
asymptote rectangle to the other, is called the conjugate axis of the hyperbola.
Given an equation of the form (5) or (6), how can we find the coordinates of the
foci without memorizing or looking up the relation b2 c2 a2? Just as with the
11-3 Hyperbola
803
ellipse, there is a simple geometric relationship in a hyperbola that enables us to get
the same result using the Pythagorean theorem. To see this relationship, we rewrite
b2 c2 a2 in the form
c2 a2 b2
(11)
Note in the figures in Theorem 1 below that the distance from the center to a focus
is the same as the distance from the center to a corner of the asymptote rectangle.
Stated in another way:
A circle, with center at the origin, that passes through all four corners
of the asymptote rectangle also passes through all foci of hyperbolas with
asymptotes determined by the diagonals of the rectangle.
We summarize all the preceding results in Theorem 1 for convenient reference.
Theorem 1
Standard Equations of a Hyperbola with Center at (0, 0)
1.
x2 y2
1
a2 b2
x intercepts: a (vertices)
y intercepts: none
Foci: F(c, 0), F(c, 0)
c2 a2 b2
y
b
c
F
c
a
F
a
c
x
b
Transverse axis length 2a
Conjugate axis length 2b
2.
y2 x2
1
a2 b2
y
x intercepts: none
y intercepts: a (vertices)
Foci: F(0, c), F(0, c)
c2 a2 b2
Transverse axis length 2a
Conjugate axis length 2b
c
F
a
c
b
b
x
a
c
F
[Note: Both graphs are symmetric with respect to the x axis, y axis, and origin.]
EXPLORE-DISCUSS 1
The line through a focus F of a hyperbola that is perpendicular to the transverse
axis intersects the hyperbola in two points G and H. For each of the two standard
equations of a hyperbola with center (0, 0), find an expression in terms of a and
b for the distance from G to H.
804
11
Additional Topics in Analytic Geometry
EXAMPLE 1
Graphing Hyperbolas
Sketch the graph of each equation, find the coordinates of the foci, and find the lengths
of the transverse and conjugate axes.
(A) 9x2 16y2 144
(B) 16y2 9x2 144
(C) 2x2 y2 10
(A) First, write the equation in standard form by dividing both sides by 144:
Solutions
9x2 16y2 144
x2
y2
1
16
9
a 2 16 and b 2 9
Locate x intercepts, x 4; there are no y intercepts. Sketch the asymptotes
using the asymptote rectangle, then sketch in the hyperbola (Fig. 5).
y
5
Foci: c2 a2 b2
c
c
6
c
F
F
16 9
x
6
25
c5
5
Thus, the foci are F(5, 0) and F(5, 0)
FIGURE 5 9x2 16y2 144.
Transverse axis length 2(4) 8
Conjugate axis length 2(3) 6
(B)
16y2 9x2 144
y2
x2
1
9
16
Locate y intercepts, y 3; there are no x intercepts. Sketch the asymptotes
using the asymptote rectangle, then sketch in the hyperbola (Fig. 6). It is important to note that the transverse axis and the foci are on the y axis.
y
6
c
a 2 9 and b 2 16
F
Foci: c2 a2 b2
c
x
6
6
9 16
25
c
F
6
FIGURE 6 16y2 9x2 144.
c5
Thus, the foci are F(0, 5) and F(0, 5).
Transverse axis length 2(3) 6
Conjugate axis length 2(4) 8
11-3 Hyperbola
2x 2 y2 10
(C)
x2
y2
1
5
10
5
Foci: c 2 a2 b2
c
c
c
F
a 2 5 and b 2 10
Locate x intercepts, x 5; there are no y intercepts. Sketch the asymptotes
using the asymptote rectangle, then sketch in the hyperbola (Fig. 7).
y
5
805
F
5 10
x
5
15
c 15
5
FIGURE 7 2x2 y2 10.
Thus, the foci are F(15, 0) and F(15, 0).
Transverse axis length 25 4.47
Conjugate axis length 210 6.32
To graph the equation 9x2 16y2 144 of Example 1A on a graphing
9x2 144
utility we first solve the equation for y, obtaining y . We then graph
16
9x2 144
each of the two functions. The graph of y is the upper half of the
16
9x2 144
hyperbola, and the graph of y is the lower half.
16
Remark.
Matched Problem 1
Sketch the graph of each equation, find the coordinates of the foci, and find the lengths
of the transverse and conjugate axes.
(A) 16x2 25y2 400
(B) 25y2 16x2 400
(C) y2 3x2 12
Hyperbolas of the form
x2 y2
1
M N
and
y2 x2
1
N M
M, N 0
are called conjugate hyperbolas. In Example 1 and Matched Problem 1, the hyperbolas in parts A and B are conjugate hyperbolas—they share the same asymptotes.
CAUTION
When making a quick sketch of a hyperbola, it is a common error to have the
hyperbola opening up and down when it should open left and right, or vice
versa. The mistake can be avoided if you first locate the intercepts accurately.
806
11
Additional Topics in Analytic Geometry
EXAMPLE 2
Finding the Equation of a Hyperbola
Find an equation of a hyperbola in the form
y2 x2
1
M N
M, N 0
if the center is at the origin, and:
(A) Length of transverse axis is 12
Length of conjugate axis is 20
Solutions
(B) Length of transverse axis is 6
Distance of foci from center is 5
(A) Start with
y2 x2
1
a2 b2
and find a and b:
a
12
6
2
and
b
20
10
2
Thus, the equation is
y2
x2
1
36 100
(B) Start with
y2 x2
1
a2 b2
and find a and b:
a
To find b, sketch the asymptote rectangle (Fig. 8), label known parts, and use
the Pythagorean theorem:
y
5
6
3
2
F
b2 52 32
5
b
3
b
16
x
b4
Thus, the equation is
5
F
FIGURE 8 Asymptote rectangle.
y2
x2
1
9
16
11-3 Hyperbola
Matched Problem 2
807
Find an equation of a hyperbola in the form
x2 y2
1
M N
M, N 0
if the center is at the origin, and:
(A) Length of transverse axis is 50
Length of conjugate axis is 30
EXPLORE-DISCUSS 2
(B) Length of conjugate axis is 12
Distance of foci from center is 9
(A) Does the line with equation y x intersect the hyperbola with equation
x2 (y2/4) 1? If so, find the coordinates of all intersection points.
(B) Does the line with equation y 3x intersect the hyperbola with equation
x2 (y2/4) 1? If so, find the coordinates of all intersection points.
(C) For which values of m does the line with equation y mx intersect the hyperx2 y2
bola 2 2 1? Find the coordinates of all intersection points.
a
b
• Applications
You may not be aware of the many important uses of hyperbolic forms. They are
encountered in the study of comets; the loran system of navigation for pleasure boats,
ships, and aircraft; sundials; capillary action; nuclear cooling towers; optical and
radiotelescopes; and contemporary architectural structures. The TWA building at
Kennedy Airport is a hyperbolic paraboloid, and the St. Louis Science Center Planetarium is a hyperboloid. (See Fig. 9.)
Comet
Ship
Sun
S3
S1
p1
Comet around sun
(a)
q2
S2
p2
q1
Loran navigation
(b)
St. Louis planetarium
(c)
FIGURE 9 Uses of hyperbolic forms.
Some comets from outer space occasionally enter the sun’s gravitational field,
follow a hyperbolic path around the sun (with the sun at a focus), and then leave,
808
11
Additional Topics in Analytic Geometry
never to be seen again [Fig. 9(a)]. In the loran system of navigation, transmitting stations in three locations, S1, S2, and S3 [Fig. 9(b)], send out signals simultaneously. A
ship with a receiver records the difference in the arrival times of the signals from S1
and S2 and also records the difference in arrival times of the signals from S2 and S3.
The difference in arrival times can be transformed into differences of the distances
that the ship is to S1 and S2 and to S2 and S3. Plotting all points so that these differences in distances remain constant produces two branches, p1 and p2, of a hyperbola
with foci S1 and S2 and two branches, q1 and q2, of a hyperbola with foci S2 and S3.
It is easy to tell which branches the ship is on by noting the arrival times of the signals from each station. The intersection of a branch from each hyperbola locates the
ship. Most of these calculations are now done by shipboard computers, and positions
in longitude and latitude are given. This system of navigation is widely used for
coastal navigation. Inexpensive loran units are now found on many small pleasure
boats. Figure 9(c) illustrates a hyperboloid used architecturally. With such structures,
thin concrete shells can span large spaces.
Answers to Matched Problems
y
1. (A)
x2
y2
1
25 16
Foci: F
(41, 0), F (41, 0)
Transverse axis length 10
Conjugate axis length 8
10
c
F
10 c
F
c
x
10
10
y
(B)
x2
y2
1
16 25
Foci: F
(0, 41), F(0, 41)
Transverse axis length 8
Conjugate axis length 10
10
c F
c
x
10
10
c F
10
y
(C)
y2
x2
1
12
4
Foci: F
(0, 4), F(0, 4)
Transverse axis length 212 6.93
Conjugate axis length 4
6
c F
c
x
5
5
c F
6
2. (A)
x2
y2
1
625 225
(B)
y2
x2
1
45 36
11-3 Hyperbola
EXERCISE
809
11-3
A
Sketch a graph of each equation in Problems 1–8, find the
coordinates of the foci, and find the lengths of the transverse
and conjugate axes.
1.
x2 y2
1
9
4
2.
y2
x2
1
9
25
3.
y2 x2
1
4
9
4.
y2
x2
1
25
9
19. (A) How many hyperbolas have center at (0, 0) and a focus
at (1, 0)? Find their equations.
(B) How many ellipses have center at (0, 0) and a focus at
(1, 0)? Find their equations.
(C) How many parabolas have vertex at (0, 0) and focus at
(1, 0)? Find their equations.
20. How many hyperbolas have the lines y 2x as asymptotes? Find their equations.
5. 4x2 y2 16
6. x2 9y2 9
In Problems 21–24, graph each system of equations in the
same rectangular coordinate system and find the coordinates
of any points of intersection.
7. 9y2 16x2 144
8. 4y2 25x2 100
Check Problems 21–24 with a graphing utility.
21. 3y2 4x2 12
y2 x2 25
22. y2 x2 3
y2 x2 5
B
23. 2x2 y2 24
x2 y2 12
24. 2x2 y2 17
x2 y2 5
Sketch a graph of each equation in Problems 9–12, find the
coordinates of the foci, and find the lengths of the transverse
and conjugate axes.
In Problems 25–28, find all points of intersection for each
system of equations to three decimal places.
9. 3x2 2y2 12
10. 3x2 4y2 24
11. 7y2 4x2 28
12. 3y2 2x2 24
In Problems 13–18, find an equation of a hyperbola in the
form
x2 y2
1
M N
or
y2 x2
1
N M
if the center is at the origin, and:
13. Transverse axis on y axis
Transverse axis length 10
Conjugate axis length 18
14. Transverse axis on x axis
Transverse axis length 22
Conjugate axis length 2
15. Transverse axis on x axis
Transverse axis length 14
Distance of foci from center 9
16. Transverse axis on y axis
Conjugate axis length 30
Distance of foci from center 25
M, N 0
Check Problems 25–28 with a graphing utility.
25. y2 x2 9
2y x 8
y0
26. y2 x2 4
yx6
y0
27. y2 x2 4
y2 2x2 36
28.
y2 x2 1
2y2 x2 16
In Problems 29–32, determine whether the statement is true
or false. If true, explain why. If false, give a counterexample.
29. The line segment joining the foci of a hyperbola has greater
length than the conjugate axis.
30. The line segment joining the foci of a hyperbola has greater
length than the transverse axis.
31. Every line through the center of 4x2 y2 16 intersects
the hyperbola in exactly two points.
32. Every nonvertical line through a vertex of 4x2 y2 16 intersects the hyperbola in exactly two points.
C
17. Conjugate axis on x axis
Conjugate axis length 12
Distance between foci 122
Eccentricity. The set of points in a plane each of whose distance from a fixed point is e times its distance from a fixed line
is a conic section. The positive number e is called the eccentricity of the conic section. Problems 33 and 34 below and
Problems 33 and 34 in Section 12-2 illustrate an approach to
defining the conic sections in terms of eccentricity.
18. Conjugate axis on y axis
Transverse axis length 2
Distance between foci 48
33. Find an equation of the set of points in a plane each of
whose distance from (3, 0) is three-halves its distance from
the line x 43. Identify the geometric figure.
810
11
Additional Topics in Analytic Geometry
34. Find an equation of the set of points in a plane each of
whose distance from (0, 4) is four-thirds its distance from
the line y 94. Identify the geometric figure.
In Problems 35–38, use a graphing utility to find the coordinates of all points of intersection to two decimal places.
35. 2x2 3y2 20, 7x 15y 10
36. y2 3x2 8, x2 y
3
37. 24y2 18x2 175, 90x2 3y2 200
38. 8x2 7y2 58, 4y2 11x2 45
Nuclear cooling tower
(a)
y
APPLICATIONS
500
39. Architecture. An architect is interested in designing a thinshelled dome in the shape of a hyperbolic paraboloid, as
shown in figure (a). Find the equation of the hyperbola located in a coordinate system [Fig. (b)] satisfying the indicated conditions. How far is the hyperbola above the vertex
6 feet to the right of the vertex? Compute the answer to two
decimal places.
x
500
500
500
Hyperbola part of dome
(b)
Hyperbola
If the tower is 500 feet tall, the top is 150 feet above the
center of the hyperbola, and the base is 350 feet below the
center, what is the radius of the top and the base? What is
the radius of the smallest circular cross section in the
tower? Compute answers to 3 significant digits.
41. Space Science. In tracking space probes to the outer planets,
NASA uses large parabolic reflectors with diameters equal to
two-thirds the length of a football field. Needless to say,
many design problems are created by the weight of these reflectors. One weight problem is solved by using a hyperbolic
reflector sharing the parabola’s focus to reflect the incoming
electromagnetic waves to the other focus of the hyperbola
where receiving equipment is installed (see the figure).
Parabola
Hyperbolic paraboloid
(a)
y
(8, 12)
Incoming
wave
10
x
10
Common
focus
F
10
Hyperbola
Hyperbola part of dome
(b)
40. Nuclear Power. A nuclear cooling tower is a hyperboloid,
that is, a hyperbola rotated around its conjugate axis, as
shown in Figure (a). The equation of the hyperbola in Figure (b) used to generate the hyperboloid is
x2
y2
1
1002 1502
Hyperbola
focus
F
Parabola
Receiving cone
(a)
11-4
Translation of Axes
811
For the receiving antenna shown in the figure, the common
focus F is located 120 feet above the vertex of the parabola,
and focus F (for the hyperbola) is 20 feet above the vertex.
The vertex of the reflecting hyperbola is 110 feet above the
vertex for the parabola. Introduce a coordinate system by
using the axis of the parabola as the y axis (up positive), and
let the x axis pass through the center of the hyperbola (right
positive). What is the equation of the reflecting hyperbola?
Write y in terms of x.
Radiotelescope
(b)
SECTION
11-4
Translation of Axes
•
•
•
•
Translation of Axes
Standard Equations of Translated Conics
Graphing Equations of the Form Ax2 Cy2 Dx Ey F 0
Finding Equations of Conics
In the last three sections we found standard equations for parabolas, ellipses, and
hyperbolas located with their axes on the coordinate axes and centered relative to the
origin. What happens if we move conics away from the origin while keeping their
axes parallel to the coordinate axes? We will show that we can obtain new standard
equations that are special cases of the equation Ax2 Cy2 Dx Ey F 0,
where A and C are not both zero. The basic mathematical tool used in this endeavor
is translation of axes. The usefulness of translation of axes is not limited to graphing
conics, however. Translation of axes can be put to good use in many other graphing
situations.
• Translation of Axes
A translation of coordinate axes occurs when the new coordinate axes have the same
direction as and are parallel to the original coordinate axes. To see how coordinates
in the original system are changed when moving to the translated system, and vice
versa, refer to Figure 1.
FIGURE 1 Translation of
coordinates.
y
y
P(x, y)
P(x
, y
)
y
y
(0
, 0
) (h, k)
0
(0, 0)
0
x
x
x
x