794
10 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
Section 10-3 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:
DEFINITION
1
HYPERBOLA
A hyperbola is the set of all points P in a 兩 d ⫺ d 兩 ⫽ Constant
1
2
plane such that the absolute value of the
difference of the distances of P to two
P
fixed points in the plane is a positive cond1
stant. Each of the fixed points, F⬘ and F,
V
is called a focus. The intersection points
V⬘
F⬘
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 V⬘V is called the
transverse axis. The midpoint of the
transverse axis is the center of the hyperbola.
d2
F
Drawing a Hyperbola
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
⫺
冢Straightedge
length 冣 冢length 冣
⫽ Constant
10-3 Hyperbola
795
FIGURE 1
Drawing a hyperbola.
A
B
String
F
F⬘
Standard Equations and Their Graphs
FIGURE 2
Hyperbola with foci on the x
axis.
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:
兩d1 ⫺ d2兩 ⬍ 2c
2a ⬍ 2c
c⬎0
兩 d1 ⫺ d2 兩 ⫽ Positive constant
a⬍c
(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
兩d1 ⫺ d2兩 ⫽ 2a
兩d(P, F⬘) ⫺ d(P, F)兩 ⫽ 2a
兩兹(x ⫹ c) ⫹ y ⫺ 兹(x ⫺ c)2 ⫹ y2兩 ⫽ 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
y
x
⫺ 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.
796
10 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
To simplify equation (3) further, we let
b2 ⫽ c2 ⫺ a2
b⬎0
(4)
to obtain
x2 y2
ⴚ ⴝ1
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 ⫽ ⫾兹⫺b2
An imaginary number
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
FIGURE 3
Hyperbola with foci on the y
axis.
y
y2 x2
ⴚ ⴝ1
a2 b2
d1
(6)
P (x, y)
where the relationship among a, b, and c remains the same as before:
F (0, c)
d2
F⬘(0, ⫺c)
c⬎0
兩 d1 ⫺ d2 兩 ⫽ Positive constant
x
b2 ⫽ c2 ⫺ a2
(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
兹
a2
b
y⫽⫾ x 1⫺ 2
a
x
(8)
As x changes so that 兩x兩 becomes larger, the expression 1 ⫺ (a2/x2) within the radical approaches 1. Hence, for large values of 兩x兩, 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.
797
10-3 Hyperbola
FIGURE 4
Asymptotes.
Asymptote
b
y⫽⫺ x
a
Asymptote
b
x
a
y
y⫽
y2
x2
⫺
⫽1
a2 b2
x
b
⫺a
a
0
⫺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 of equations (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 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
y
x intercepts: ⫾a (vertices)
y intercepts: none
Foci: F⬘(⫺c, 0), F(c, 0)
b
⫺c
c ⫽a ⫹b
2
2
c
F⬘
⫺a
2
Transverse axis length ⫽ 2a
Conjugate axis length ⫽ 2b
F
a
⫺b
c
x
798
10 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
2.
THEOREM
y2 x2
⫺ ⫽1
a2 b2
y
x intercepts: none
1
c
y intercepts: ⫾a (vertices)
continued
Foci: F⬘(0, ⫺c), F(0, c)
F
a
c
⫺b
b
x
⫺a
c2 ⫽ a2 ⫹ b2
⫺c
F⬘
Transverse axis length ⫽ 2a
Conjugate axis length ⫽ 2b
[Note: Both graphs are symmetric with respect to the x axis, y axis, and
origin.]
Explore/Discuss
1
EXAMPLE
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.
Graphing Hyperbolas
1
Sketch the graph of each equation, find the coordinates of the foci, and find
the lengths of the transverse and conjugate axes. Check by graphing on a
graphing utility.
(B) 16y2 ⫺ 9x2 ⫽ 144
(C) 2x2 ⫺ y2 ⫽ 10
(A) 9x2 ⫺ 16y2 ⫽ 144
(A) First, write the equation in standard form by dividing both sides by 144:
Solutions
9x2 ⫺ 16y2 ⫽ 144
y2
x2
⫺ ⫽1
16
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).
FIGURE 5
9x2 ⫺ 16y2 ⫽ 144.
y
Foci: c2 ⫽ a2 ⫹ b2
5
⫽ 16 ⫹ 9
⫽ 25
c
⫺c
⫺6
c
F⬘
a2 ⫽ 16 and b2 ⫽ 9
F
x
6
c⫽5
Thus, the foci are F⬘(⫺5, 0) and F(5, 0).
⫺5
Transverse axis length ⫽ 2(4) ⫽ 8
Conjugate axis length ⫽ 2(3) ⫽ 6
10-3 Hyperbola
799
To check the graph on a graphing utility, we solve the original equation
for y:
FIGURE 6
y1 ⫽ 兹(9x2 ⫺ 144)/16;
y2 ⫽ ⫺兹(9x2 ⫺ 144)/16.
9x2 ⫺ 16y2 ⫽ 144
6
y2 ⫽ (9x2 ⫺ 144)/16
⫺9
y ⫽ ⫾兹(9x2 ⫺ 144)/16
9
This produces two functions whose graphs are shown in Figure 6.
16y2 ⫺ 9x2 ⫽ 144
(B)
⫺6
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. 7). It
is important to note that the transverse axis and the foci are on the y axis.
FIGURE 7
16y2 ⫺ 9x2 ⫽ 144.
y
Foci: c2 ⫽ a2 ⫹ b2
6
c
F
⫽ 9 ⫹ 16
c
⫽ 25
x
⫺6
c⫽5
6
⫺c
a2 ⫽ 9 and b2 ⫽ 16
Thus, the foci are F⬘(0, ⫺5) and F(0, 5).
F⬘
⫺6
Transverse axis length ⫽ 2(3) ⫽ 6
Conjugate axis length ⫽ 2(4) ⫽ 8
A check of the graph is shown in Figure 8.
FIGURE 8
6
y1 ⫽ 兹(9x2 ⫹ 144)/16;
y2 ⫽ ⫺兹(9x2 ⫹ 144)/16.
⫺9
9
⫺6
(C)
FIGURE 9
2x2 ⫺ y2 ⫽ 10
x2
y2
⫺
⫽1
5
10
2x2 ⫺ y2 ⫽ 10.
y
a2 ⫽ 5 and b2 ⫽ 10
Locate x intercepts, x ⫽ ⫾兹5; there are no y intercepts. Sketch the asymptotes using the asymptote rectangle, then sketch in the hyperbola (Fig. 9).
5
c
⫺c
⫺5
c
F⬘
F
⫺5
x
5
Foci: c2 ⫽ a2 ⫹ b2
⫽ 5 ⫹ 10
⫽ 15
c ⫽ 兹15
800
10 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
Thus, the foci are F⬘(⫺兹15, 0) and F(兹15, 0).
FIGURE 10
y1 ⫽ 兹2x2 ⫺ 10;
y2 ⫽ ⫺兹2x2 ⫺ 10.
Transverse axis length ⫽ 2兹5 艐 4.47
Conjugate axis length ⫽ 2兹10 艐 6.32
6
A check of the graph is shown in Figure 10.
⫺9
9
⫺6
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. Check by graphing on a graphing
utility.
(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
EXAMPLE
2
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.
Finding the Equation of a Hyperbola
Find an equation of a hyperbola in the form
y 2 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
(A) Start with
y2 x2
⫺ ⫽1
a2 b2
(B) Length of transverse axis is 6
Distance of foci from center is 5
10-3 Hyperbola
801
and find a and b:
a⫽
12
⫽6
2
and
b⫽
20
⫽ 10
2
Thus, the equation is
x2
y2
⫺
⫽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. 11), label known parts, and
use the Pythagorean theorem:
FIGURE 11
Asymptote rectangle.
y
5
b2 ⫽ 52 ⫺ 32
F
⫽ 16
5
⫺b
b⫽4
3
b
⫺5
6
⫽3
2
x
Thus, the equation is
y2
x2
⫺
⫽1
9
16
F⬘
MATCHED PROBLEM
2
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.
802
10 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
Explore/Discuss
2
continued
(B) Does the line with equation y ⫽ 3x intersect the hyperbola with
equation x2 ⫺ (y 2/4) ⫽ 1? If so, find the coordinates of all intersection points.
(C) For which values of m does the line with equation y ⫽ mx intersect
x2 y2
the hyperbola 2 ⫺ 2 ⫽ 1? Find the coordinates of all intersection
a
b
points.
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. With such structures, thin concrete
shells can span large spaces [see Fig. 12(a)]. Some comets from outer space occasionally enter the sun’s gravitational field, follow a hyperbolic path around the
sun (with the sun as a focus), and then leave, never to be seen again [Fig. 12(b)].
The next example illustrates the use of hyperbolas in navigation.
FIGURE 12
Uses of hyperbolic forms.
Comet
Sun
St. Louis Planetarium
(a)
EXAMPLE
3
Comet around sun
(b)
Navigation
A ship is traveling on a course parallel to and 60 miles from a straight shoreline. Two transmitting stations, S1 and S2, are located 200 miles apart on the
shoreline (see Fig. 13). By timing radio signals from the stations, the ship’s
navigator determines that the ship is between the two stations and 50 miles
closer to S2 than to S1. Find the distance from the ship to each station. Round
answers to one decimal place.
FIGURE 13
d1 ⫺ d2 ⫽ 50.
d1
60 miles
d2
S2
S1
200 miles
10-3 Hyperbola
803
If d1 and d2 are the distances from the ship to S1 and S2, respectively, then
d1 – d2 ⫽ 50 and the ship must be on the hyperbola with foci at S1 and S2 and
fixed difference 50, as illustrated in Figure 14. In the derivation of the equation
of a hyperbola, we represented the fixed difference as 2a. Thus, for the hyperbola
in Figure 14 we have:
Solution
FIGURE 14
y
200
c ⫽ 100
S1
(x, 60)
S2
⫺100
100
a ⫽ 12(50) ⫽ 25
x
b ⫽ 兹1002 ⫺ 252 ⫽ 兹9,375
The equation for this hyperbola is
y2
x2
⫺
⫽1
625 9,375
Substitute y ⫽ 60 and solve for x (see Fig. 14):
x2
602
⫺
⫽1
625 9,375
x2
3,600
⫽
⫹1
625 9,375
x2 ⫽ 625
3,600 ⫹ 9,375
9,375
⫽ 865
Thus, x ⫽ 兹865 艐 29.41. (The negative square root is discarded, since the ship
is closer to S2 than to S1.)
Distance from ship to S1
d1 ⫽ 兹(29.41 ⫹ 100)2 ⫹ 602
Distance from ship to S2
d2 ⫽ 兹(29.41 ⫺ 100)2 ⫹ 602
⫽ 兹20,346.9841
⫽ 兹8,582.9841
⬇ 142.6
⬇ 92.6 miles
Notice that the difference between these two distances is 50, as it should be.
MATCHED PROBLEM
Repeat Example 3 if the ship is 80 miles closer to S2 than to S1.
3
Example 3 illustrates a simplified form of the loran (LOng RAnge Navigation) system. In practice, three transmitting stations are used to send out signals
simultaneously (Fig. 15), instead of the two used in Example 3. A computer
onboard a ship will record these signals and use them to determine the differences
of the distances that the ship is to S1 and S2, and to S2 and S3.
804
10 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
FIGURE 15
Loran navigation.
Ship
S3
q2
S1
S2
p1
p2
q1
Plotting all points so that these 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
comparing the signals from each station. The intersection of a branch of each
hyperbola locates the ship and the computer expresses this in terms of longitude
and latitude.
Answers to Matched Problems
1. (A)
y
x2
y2
⫺
⫽1
25 16
Foci: F ⬘(⫺兹41, 0), F (兹41, 0)
Transverse axis length ⫽ 10
Conjugate axis length ⫽ 8
10
c
F⬘
⫺10 ⫺c
F
c
x
10
⫺10
(B)
y
x2
y2
⫺
⫽1
16 25
Foci: F ⬘(0, ⫺兹41), F (0, 兹41)
Transverse axis length ⫽ 8
Conjugate axis length ⫽ 10
10
c
F
c
x
⫺10
10
⫺c F⬘
⫺10
(C)
y
x2
y2
⫺
⫽1
12
4
Foci: F ⬘(0, ⫺4), F (0, 4)
Transverse axis length ⫽ 2兹12 ⬇ 6.93
Conjugate axis length ⫽ 4
6
c F
c
x
⫺5
5
⫺c F⬘
⫺6
10-3 Hyperbola
x2
y2
⫺
⫽1
625 225
2. (A)
(B)
x2
y2
⫺
⫽1
45 36
805
3. d1 ⫽ 159.5 miles, d2 ⫽ 79.5 miles
EXERCISE 10-3
coordinates of the foci, and find the lengths of the transverse
and conjugate axes. Check by graphing on a graphing utility.
A
13. 3x2 ⫺ 2y2 ⫽ 12
14. 3x2 ⫺ 4y2 ⫽ 24
15. 7y2 ⫺ 4x2 ⫽ 28
16. 3y2 ⫺ 2x2 ⫽ 24
In Problems 1–4, match each equation with one of graphs
(a)–(d).
1. x2 ⫺ y2 ⫽ 1
2. y2 ⫺ x2 ⫽ 1
3. y2 ⫺ x2 ⫽ 4
4. x2 ⫺ y2 ⫽ 4
y
In Problems 17–28, find an equation of a hyperbola in the
form
x2 y2
⫺ ⫽1
M N
y
5
y2 x2
⫺ ⫽1
N M
or
if the center is at the origin, and
5
17. The graph is
y
x
⫺5
5
x
⫺5
10
5
(5, 4)
⫺5
⫺5
(a)
x
⫺10
10
(b)
y
y
5
⫺10
5
18. The graph is
x
⫺5
5
y
x
⫺5
5
10
⫺5
(4, 5)
⫺5
(c)
(d)
Sketch a graph of each equation in Problems 5–12, find the
coordinates of the foci, and find the lengths of the transverse
and conjugate axes. Check by graphing on a graphing utility.
5.
x2 y2
⫺ ⫽1
9
4
y2 x2
7.
⫺ ⫽1
4
9
9. 4x2 ⫺ y2 ⫽ 16
11. 9y ⫺ 16x ⫽ 144
2
2
6.
x2
y2
⫺
⫽1
9
25
x
⫺10
10
⫺10
19. The graph is
y
y2
x2
8.
⫺ ⫽1
25
9
10
10. x2 ⫺ 9y2 ⫽ 9
12. 4y ⫺ 25x ⫽ 100
2
(3, 5)
x
2
⫺10
10
B
⫺10
Sketch a graph of each equation in Problems 13–16, find the
M, N ⬎ 0
806
10 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY
20. The graph is
y
10
(5, 3)
35. y2 ⫺ x2 ⫽ 9
2y ⫺ x ⫽ 8
36. y2 ⫺ x2 ⫽ 4
y⫺x⫽6
37. y2 ⫺ x2 ⫽ 4
y2 ⫹ 2x2 ⫽ 36
38. y2 ⫺ x2 ⫽ 1
2y2 ⫹ x2 ⫽ 16
C
x
⫺10
10
⫺10
21. Transverse axis on x axis
Transverse axis length ⫽ 14
Conjugate axis length ⫽ 10
22. Transverse axis on x axis
Transverse axis length ⫽ 8
Conjugate axis length ⫽ 6
23. Transverse axis on y axis
Transverse axis length ⫽ 24
Conjugate axis length ⫽ 18
24. Transverse axis on y axis
Transverse axis length ⫽ 16
Conjugate axis length ⫽ 22
25. Transverse axis on x axis
Transverse axis length ⫽ 18
Distance of foci from center ⫽ 11
26. Transverse axis on x axis
Transverse axis length ⫽ 16
Distance of foci from center ⫽ 10
27. Conjugate axis on x axis
Conjugate axis length ⫽ 14
Distance of foci from center ⫽ 兹200
28. Conjugate axis on x axis
Conjugate axis length ⫽ 10
Distance of foci from center ⫽ 兹70
29. (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 center at (0, 0) and focus
at (1, 0)? Find their equations.
Eccentricity. Problems 39 and 40 below and Problems 39 and
40 in Exercise 10-2 are related to a property of conics called
eccentricity, which is denoted by a positive real number E.
Parabolas, ellipses, and hyperbolas all can be defined in terms
of E, a fixed point called a focus, and a fixed line not
containing the focus called a directrix as follows: 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 an ellipse if 0 ⬍ E ⬍ 1, a
parabola if E ⫽ 1, and a hyperbola if E ⬎ 1.
39. 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.
40. 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 41–44, find the coordinates of all points of
intersection to two decimal places.
41. 2x2 ⫺ 3y2 ⫽ 20, 7x ⫹ 15y ⫽ 10
42. y2 ⫺ 3x2 ⫽ 8, x2 ⫽ ⫺
y
3
43. 24y2 ⫺ 18x2 ⫽ 175, 90x2 ⫹ 3y2 ⫽ 200
44. 8x2 ⫺ 7y2 ⫽ 58, 4y2 ⫺ 11x2 ⫽ 45
APPLICATIONS
45. Architecture. An architect is interested in designing a
thin-shelled 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.
Hyperbola
30. How many hyperbolas have the lines y ⫽ ⫾2x as asymptotes? Find their equations.
In Problems 31–38, find the coordinates of all points of intersection. Round any approximate values to three decimal places.
31. 3y2 ⫺ 4x2 ⫽ 12
y2 ⫹ x2 ⫽ 25
32. y2 ⫺ x2 ⫽ 3
y2 ⫹ x2 ⫽ 5
33. 2x2 ⫹ y2 ⫽ 24
x2 ⫺ y2 ⫽ ⫺12
34. 2x2 ⫹ y2 ⫽ 17
x2 ⫺ y2 ⫽ ⫺5
Parabola
Hyperbolic paraboloid
(a)
10-3 Hyperbola
y
(8, 12)
10
x
⫺10
10
47. 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).
Hyperbola part of dome
(b)
46. 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
807
Incoming
wave
Common
focus
F
Hyperbola
x2
y2
⫺
⫽1
1002 1502
Hyperbola
focus
F⬘
Parabola
Receiving cone
(a)
Nuclear cooling tower
(a)
Radiotelescope
y
500
x
⫺500
500
⫺500
Hyperbola part of dome
(b)
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.
(b)
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.