C. The Foci of an Ellipse
Tải bản đầy đủ - 0trang
cob19545_ch08_716-730.qxd
10/25/10
2:44 PM
Page 721
College Algebra G&M—
8–15
721
Section 8.2 The Circle and the Ellipse
Figure 8.17
y
(0, b)
P(x, y)
(a, 0)
x
(Ϫa, 0)
(Ϫc, 0)
(c, 0)
calculating ease we use a central ellipse). Note the vertices have coordinates 1Ϫa, 02 and
(a, 0), and the endpoints of the minor axis have coordinates 10, Ϫb2 and (0, b) as
before. It is customary to assign foci the coordinates f1 S 1Ϫc, 02 and f2 S 1c, 02. We
can calculate the distance between (c, 0) and any point P(x, y) on the ellipse using the
distance formula:
21x Ϫ c2 2 ϩ 1y Ϫ 02 2
Likewise the distance between 1Ϫc, 02 and any point (x, y) is
21x ϩ c2 2 ϩ 1y Ϫ 02 2
(0, Ϫb)
According to the definition, the sum must be constant:
21x Ϫ c2 2 ϩ y2 ϩ 21x ϩ c2 2 ϩ y2 ϭ k
EXAMPLE 6
ᮣ
Finding the Value of k from the Definition of an Ellipse
Use the definition of an ellipse and the diagram given to determine the constant k
used for this ellipse (also see the following Worthy of Note). Note that
a ϭ 5, b ϭ 3, and c ϭ 4.
y
(0, 3)
P(3, 2.4)
(Ϫ5, 0)
(Ϫ4, 0)
(4, 0)
(5, 0)
x
(0, Ϫ3)
Solution
ᮣ
21x Ϫ c2 2 ϩ 1y Ϫ 02 2 ϩ 21x ϩ c2 2 ϩ 1y Ϫ 02 2 ϭ k
213 Ϫ 42 ϩ 12.4 Ϫ 02 ϩ 213 ϩ 42 ϩ 12.4 Ϫ 02 ϭ k
2
WORTHY OF NOTE
Note that if the foci are coincident
(both at the origin) the “ellipse” will
k
actually be a circle with radius ;
2
2x2 ϩ y2 ϩ 2x2 ϩ y2 ϭ k leads to
k2
x2 ϩ y2 ϭ . In Example 6 we
4
10
ϭ 5, and if
found k ϭ 10, giving
2
we used the “string” to draw the
circle, the pencil would be 5 units
from the center, creating a circle of
radius 5.
2
2
2
21Ϫ12 ϩ 2.4 ϩ 27 ϩ 2.4 ϭ k
16.76 ϩ 154.76 ϭ k
2.6 ϩ 7.4 ϭ k
10 ϭ k
The constant value for this ellipse is 10 units.
2
2
2
2
given
substitute
add
simplify radicals
compute square roots
result
Now try Exercises 45 through 48
In Example 6, the sum of the distances
could also be found by moving the point (x, y)
to the location of a vertex (a, 0), then using
the symmetry of the ellipse. The sum is identical to the length of the major axis, since the
overlapping part of the string from (c, 0) to
(a, 0) is the same length as from (Ϫa, 0) to
(Ϫc, 0) (see Figure 8.18). This shows the
constant k is equal to 2a regardless of the distance between foci.
As we noted, the result is
ᮣ
Figure 8.18
y
d1 ϩ d2 ϭ 2a
d1
d2
(Ϫa, 0)
(Ϫc, 0)
21x Ϫ c2 2 ϩ y2 ϩ 21x ϩ c2 2 ϩ y2 ϭ 2a
(c, 0)
(a, 0)
x
These two segments
are equal
substitute 2a for k
cob19545_ch08_716-730.qxd
12/15/10
10:51 AM
Page 722
College Algebra G&M—
722
8–16
CHAPTER 8 Analytic Geometry and the Conic Sections
The details for simplifying this expression are given in Appendix V, and the result
is very close to the standard form seen previously:
y2
x2
ϩ
ϭ1
a2
a2 Ϫ c2
y2
y2
x2
x2
ϩ
ϭ
1
ϩ
ϭ 1, we might
with
a2
b2
a2
a2 Ϫ c 2
suspect that b2 ϭ a2 Ϫ c2, and this is indeed the case. Note from Example 6 the relationship yields
By comparing the standard form
b2 ϭ a2 Ϫ c2
3 2 ϭ 52 Ϫ 42
9 ϭ 25 Ϫ 16
Additionally, when we consider that (0, b) is
Figure 8.19
a point on the ellipse, the distance from (0, b) to
y
(c, 0) must be equal to a due to symmetry (the
(0, b)
“constant distance” used to form the ellipse is always 2a). We then see in Figure 8.19, that
a
a
b2 ϩ c2 ϭ a2 (Pythagorean Theorem), yielding
b
(a, 0)
(Ϫa, 0)
2
2
2
b ϭ a Ϫ c as above.
x
(Ϫc, 0)
(c, 0)
With this development, we now have the
ability to locate the foci of any ellipse —an important step toward using the ellipse in practical
(0, Ϫb)
applications. Because we’re often asked to find
the location of the foci, it’s best to rewrite the relationship in terms of c2, using absolute value bars to allow for a major axis that is vertical: c2 ϭ Ϳa2 Ϫ b2Ϳ.
EXAMPLE 7
ᮣ
Completing the Square to Graph an Ellipse and Locate the Foci
For the ellipse defined by 25x2 ϩ 9y2 Ϫ 100x Ϫ 54y Ϫ 44 ϭ 0, find the
coordinates of the center, vertices, foci, and endpoints of the minor axis. Then
sketch the graph.
Solution
ᮣ
25x2 ϩ 9y2 Ϫ 100x Ϫ 54y Ϫ 44 ϭ 0
25x2 Ϫ 100x ϩ 9y2 Ϫ 54y ϭ 44
2
251x Ϫ 4x ϩ __ 2 ϩ 91y2 Ϫ 6y ϩ __ 2 ϭ 44
251x2 Ϫ 4x ϩ 42 ϩ 91y2 Ϫ 6y ϩ 92 ϭ 44 ϩ 100 ϩ 81
c
c
adds 25142 ϭ 100
c
adds 9192 ϭ 81
c
given
group terms; add 44
factor out lead coefficients
add 100 ϩ 81 to right-hand side
251x Ϫ 22 2 ϩ 91y Ϫ 32 2 ϭ 225
91y Ϫ 32 2
251x Ϫ 22 2
225
ϩ
ϭ
225
225
225
1x Ϫ 22 2
1y Ϫ 32 2
ϩ
ϭ1
9
25
2
2
1x Ϫ 22
1y Ϫ 32
ϩ
ϭ1
32
52
factored form
divide by 225
simplify (standard form)
write denominators
in squared form
The result shows a vertical ellipse with a ϭ 3 and b ϭ 5. The center of the ellipse
is at (2, 3). The vertices are a vertical distance of b ϭ 5 units from center at (2, 8)
and (2, Ϫ2). The endpoints of the minor axis are a horizontal distance of a ϭ 3
units from center at (Ϫ1, 3) and (5, 3). To locate the foci, we use the foci formula
cob19545_ch08_716-730.qxd
12/15/10
10:51 AM
Page 723
College Algebra G&M—
8–17
Section 8.2 The Circle and the Ellipse
723
for an ellipse: c2 ϭ Ϳa2 Ϫ b2Ϳ, giving c2 ϭ Ϳ32 Ϫ 52Ϳ ϭ 16. This shows the foci “ ”
are located a vertical distance of 4 units from center at (2, 7) and (2, Ϫ1).
y (2, 8)
Vertical ellipse
Center at (2, 3)
(2, 7)
(Ϫ1, 3)
(2, 3)
(2, Ϫ1)
(2, Ϫ2)
Endpoints of major axis (vertices)
(2, 8) and (2, Ϫ2)
(5, 3)
x
Endpoints of minor axis
(Ϫ1, 3) and (5, 3)
Location of foci
(2, 7) and (2, Ϫ1)
Length of major axis: 2b ϭ 2(5) ϭ 10
Length of minor axis: 2a ϭ 2(3) ϭ 6
Now try Exercises 49 through 54
ᮣ
For an ellipse, a focal chord is a line segment perpendicular to the major axis,
through a focus and with endpoints on the ellipse. In the Exercise Set, you are asked
2m2
to verify that the focal chord of an ellipse has length L ϭ
, where m is the length
n
of the semiminor axis and n is the length of the semimajor axis. This means the
m2
distance from the foci to the graph (along a focal chord) is , a fact can often be used
n
to help graph an ellipse. For Example 7,
m ϭ 3 and n ϭ 5, so the horizontal distance
Figure 8.20
from focus to graph (in either direction) is
9.2
9
32
ϭ . From the upper focus (2, 7), we can
5
5
now graph the additional points 12 Ϫ 1.8, 72 ϭ
11.4
10.2, 72 and (2 ϩ 1.8, 7) ϭ 13.8, 72, and Ϫ7.4
from the lower focus 12, Ϫ12 we obtain
10.2, Ϫ12 and 13.8, Ϫ12 without having to
evaluate the original equation. Graphical verϪ3.2
ification is provided in Figure 8.20. Also see
Exercises 83 and 85.
For future reference, remember the foci of an ellipse always occur on the major
axis, with a 7 c and a2 7 c2 for a horizontal ellipse, with b 7 c and b2 7 c2 for a
vertical ellipse. This makes it easier to remember the foci formula for ellipses:
c2 ϭ Ϳa2 Ϫ b2Ϳ. If any two of the values for a, b, and c are known, the relationship
between them can be used to construct the equation of the ellipse.
EXAMPLE 8
ᮣ
Finding the Equation of an Ellipse
Find the equation of the ellipse (in standard form) that has foci at (0, Ϫ2) and (0, 2),
with a minor axis 6 units in length. Then graph the ellipse
a. By hand.
b. On a graphing calculator.
m2
c. Find the distance from foci to graph along a focal chord ausing
b, and use
n
the result to verify that the endpoints of both focal chords are all on the
graph.
cob19545_ch08_716-730.qxd
10/25/10
2:45 PM
Page 724
College Algebra G&M—
724
8–18
CHAPTER 8 Analytic Geometry and the Conic Sections
Solution
ᮣ
LOOKING AHEAD
Since the foci are on the y-axis and an equal distance from (0, 0), we know this is a
vertical and central ellipse with c ϭ 2 and c2 ϭ 4. The minor axis has a length of
2a ϭ 6 units, meaning a ϭ 3 and a2 ϭ 9. To find b2, use the foci equation and solve.
Figure 8.21
foci equation (ellipse)
c2 ϭ Ϳa2 Ϫ b2Ϳ
4 ϭ Ϳ9 Ϫ b2Ϳ
Ϫ4 ϭ 9 Ϫ b2
4 ϭ 9 Ϫ b2
b2 ϭ 13
b2 ϭ 5
For the hyperbola, we’ll find that
c 7 a, and the formula for the foci
of a hyperbola will be c2 ϭ a2 ϩ b2.
y
substitute
(0, √13)
solve the absolute value equation
result
2
Since we know b must be greater than a2 (the major
axis is always longer), b2 ϭ 5 can be discarded. The
(0, 2)
(Ϫ3, 0)
2
y
x2
ϩ
ϭ 1.
2
3
1 2132 2
a. The graph is shown in Figure 8.21.
b. For a calculator generated graph, begin by solving for y.
(3, 0)
(0, Ϫ2)
x
standard form is
(0, Ϫ√13)
y2
x2
ϩ
ϭ1
original equation
9
13
13x2 ϩ 9y2 ϭ 117
clear denominators
isolate y-term
9y2 ϭ 117 Ϫ 13x2
2
117 Ϫ 13x
y2 ϭ
divide by 9
9.4
9
117 Ϫ 13x2
yϭϮ
take square roots
B
9
117 Ϫ 13X2
117 Ϫ 13X2
Y1 ϭ ϩ
, Y2 ϭ Ϫ
B
9
B
9
The graph is shown in Figure 8.22.
c. From the discussion prior to Example 8, the horizontal distance from foci to graph
m2
9
must be
. Using the TRACE feature and entering x ϭ 9/ 213 verifies
ϭ
n
213
Figure 8.22
6.2
Ϫ9.4
Ϫ6.2
Figure 8.23
6.2
Ϫ9.4
9.4
that a
9
, 2b is a point on the graph (Figure 8.23), and that aϪ
9
, 2b,
213
213
9
9
aϪ
, Ϫ2b, and a
, Ϫ2b must also be on the graph due to symmetry.
213
213
Ϫ6.2
C. You’ve just seen how we
can locate the foci of an ellipse
and use the foci and other
features to write the equation
Now try Exercises 55 through 62
ᮣ
D. Applications Involving Foci
Applications involving the foci of a conic section can take various forms. In many
cases, only partial information about the conic section is available and the ideas from
Example 8 must be used to “fill in the gaps.” In other applications, we must rewrite a
known or given equation to find information related to the values of a, b, and c.
EXAMPLE 9
ᮣ
Solving Applications Using the Characteristics of an Ellipse
In Washington, D.C., there is a park called the Ellipse located between the White
House and the Washington Monument. The park is surrounded by a path that forms
an ellipse with the length of the major axis being about 1502 ft and the minor axis
having a length of 1280 ft. Suppose the park manager wants to install water
fountains at the location of the foci. Find the distance between the fountains
rounded to the nearest foot.
cob19545_ch08_716-730.qxd
10/25/10
2:45 PM
Page 725
College Algebra G&M—
8–19
Section 8.2 The Circle and the Ellipse
Solution
ᮣ
725
Since the major axis has length 2a ϭ 1502, we know a ϭ 751 and a2 ϭ 564,001.
The minor axis has length 2b ϭ 1280, meaning b ϭ 640 and b2 ϭ 409,600. To find
c, use the foci equation:
c2 ϭ a2 Ϫ b2
ϭ 564,001 Ϫ 409,600
ϭ 154,401
c Ϸ Ϫ393 and c Ϸ 393
since we know a 7 b
substitute
subtract
square root property
The distance between the water fountains would be 213932 ϭ 786 ft.
D. You’ve just seen how
we can solve applications
involving the foci
Now try Exercises 65 through 80
ᮣ
8.2 EXERCISES
ᮣ
CONCEPTS AND VOCABULARY
Fill in the blank with the appropriate word or phrase. Carefully reread the section if needed.
1. For an ellipse, the relationship between a, b, and c
is given by the foci equation
, since
c 6 a or c 6 b.
2. The greatest distance across an ellipse is called the
and the endpoints are called
.
3. For a vertical ellipse, the length of the minor axis is
and the length of the major axis is
.
4. To write the equation 2x2 ϩ y2 Ϫ 6x ϭ 7 in
standard form,
the
in x.
5. Explain/Discuss how the relations a 7 b, a ϭ b
and a 6 b affect the graph of a conic section with
6. Suppose foci are located at (Ϫ3, 2) and (5, 2).
Discuss/Explain the conditions necessary for the
graph to be an ellipse.
equation
ᮣ
1x Ϫ h2 2
a2
ϩ
1y Ϫ k2 2
b2
ϭ 1.
DEVELOPING YOUR SKILLS
Find an equation of the circle satisfying the conditions
given, then graph the result on a graphing calculator
and locate two additional points on the graph.
15. x2 ϩ y2 Ϫ 4x ϩ 10y ϩ 4 ϭ 0
16. x2 ϩ y2 ϩ 4x ϩ 6y Ϫ 3 ϭ 0
7. center (0, 0), radius 7
17. x2 ϩ y2 ϩ 6x Ϫ 5 ϭ 0
8. center (0, 0), radius 9
18. x2 ϩ y2 Ϫ 8y Ϫ 5 ϭ 0
9. center (5, 0), radius 13
Sketch the graph of each ellipse.
19.
1x Ϫ 12 2
20.
1x Ϫ 32 2
21.
1x Ϫ 22 2
22.
1x ϩ 52 2
10. center (0, 4), radius 15
11. diameter has endpoints (4, 9) and (Ϫ2, 1)
12. diameter has endpoints (Ϫ2, Ϫ32 , and (3, 9)
Write each equation in standard form to identify the
center and radius of the circle, then sketch its graph.
13. x2 ϩ y2 Ϫ 12x Ϫ 10y ϩ 52 ϭ 0
14. x2 ϩ y2 ϩ 8x Ϫ 6y Ϫ 11 ϭ 0
9
4
25
1
ϩ
1y Ϫ 22 2
ϩ
1y Ϫ 12 2
ϩ
1y ϩ 32 2
ϩ
1y Ϫ 22 2
16
25
4
16
ϭ1
ϭ1
ϭ1
ϭ1
cob19545_ch08_716-730.qxd
12/15/10
10:52 AM
Page 726
College Algebra G&M—
726
8–20
CHAPTER 8 Analytic Geometry and the Conic Sections
23.
1x ϩ 12 2
24.
1x ϩ 12
16
ϩ
1y ϩ 22 2
ϩ
1y ϩ 32
2
36
47.
ϭ1
9
48.
(0, b) y
(0, 8)
(0, b) y
(4.8, 6)
(0, 28)
2
ϭ1
9
(6, 0) x
(Ϫ6, 0)
(96, 0) x
(Ϫ96, 0)
(0, Ϫ28 )
For each exercise, (a) write the equation in standard form,
then identify the center and the values of a and b, (b) state
the coordinates of the vertices and the coordinates of the
endpoints of the minor axis, (c) sketch the graph, and
(d) for 25–28 (only) graph the relations on a graphing
calculator and identify four additional points on the graph
whose coordinates are rational.
(0, Ϫ8)
(76.8, Ϫ60)
(0, Ϫb)
(0, Ϫb)
Find the coordinates of the (a) center, (b) vertices,
(c) foci, and (d) endpoints of the minor axis. Then
(e) sketch the graph.
49. 4x2 ϩ 25y2 Ϫ 16x Ϫ 50y Ϫ 59 ϭ 0
25. x2 ϩ 4y2 ϭ 16
26. 9x2 ϩ y2 ϭ 36
50. 9x2 ϩ 16y2 Ϫ 54x Ϫ 64y ϩ 1 ϭ 0
27. 16x2 ϩ 9y2 ϭ 144
28. 25x2 ϩ 9y2 ϭ 225
51. 25x2 ϩ 16y2 Ϫ 200x ϩ 96y ϩ 144 ϭ 0
29. 2x2 ϩ 5y2 ϭ 10
30. 3x2 ϩ 7y2 ϭ 21
52. 49x2 ϩ 4y2 ϩ 196x Ϫ 40y ϩ 100 ϭ 0
53. 6x2 ϩ 24x ϩ 9y2 ϩ 36y ϩ 6 ϭ 0
Identify each equation as that of an ellipse or circle,
then sketch its graph.
54. 5x2 Ϫ 50x ϩ 2y2 Ϫ 12y ϩ 93 ϭ 0
31. 1x ϩ 12 ϩ 41y Ϫ 22 ϭ 16
2
2
32. 91x Ϫ 22 2 ϩ 1y ϩ 32 2 ϭ 36
Find the equation of the ellipse (in standard form) that
satisfies the following conditions. Then (a) graph the
ellipse by hand, (b) confirm your graph by graphing the
ellipse on a graphing calculator, and (c) find the length
of the focal chords and verify the endpoints of the
chords are on the graph.
33. 21x Ϫ 22 2 ϩ 21y ϩ 42 2 ϭ 18
34. 1x Ϫ 62 2 ϩ y2 ϭ 49
35. 41x Ϫ 12 2 ϩ 91y Ϫ 42 2 ϭ 36
36. 251x Ϫ 32 2 ϩ 41y ϩ 22 2 ϭ 100
Complete the square in both x and y to write each
equation in standard form. Then draw a complete graph
of the relation and identify all important features,
including the domain and range.
37. 4x2 ϩ y2 ϩ 6y ϩ 5 ϭ 0
55. vertices at (Ϫ6, 0) and (6, 0);
foci at (Ϫ4, 0) and (4, 0)
56. vertices at (Ϫ8, 0) and (8, 0);
foci at (Ϫ5, 0) and (5, 0)
57. foci at (3, Ϫ6) and (3, 2);
length of minor axis: 6 units
58. foci at (Ϫ4, Ϫ3) and (8, Ϫ3);
length of minor axis: 8 units
38. x2 ϩ 3y2 ϩ 8x ϩ 7 ϭ 0
39. x2 ϩ 4y2 Ϫ 8y ϩ 4x Ϫ 8 ϭ 0
Use the characteristics of an ellipse and the graph given to
write the related equation and find the location of the foci.
40. 3x2 ϩ y2 Ϫ 8y ϩ 12x Ϫ 8 ϭ 0
41. 5x2 ϩ 2y2 ϩ 20y Ϫ 30x ϩ 75 ϭ 0
59.
60.
y
y
42. 4x ϩ 9y Ϫ 16x ϩ 18y Ϫ 11 ϭ 0
2
2
43. 2x2 ϩ 5y2 Ϫ 12x ϩ 20y Ϫ 12 ϭ 0
44. 6x2 ϩ 3y2 Ϫ 24x ϩ 18y Ϫ 3 ϭ 0
x
x
Use the definition of an ellipse to find the constant k for
each ellipse (figures are not drawn to scale).
45.
46.
y
(0, 8)
(6, 6.4)
(Ϫa, 0)
(Ϫ6, 0)
(0, Ϫ8)
61.
y
62.
y
y
(0, 12)
(6, 0)
(Ϫ9, 9.6)
(a, 0) (Ϫa, 0)
x
(Ϫ9, 0)
(0, Ϫ12)
(9, 0)
(a, 0)
x
x
x
cob19545_ch08_716-730.qxd
12/15/10
10:52 AM
Page 727
College Algebra G&M—
8–21
ᮣ
Section 8.2 The Circle and the Ellipse
WORKING WITH FORMULAS
63. Area of an Ellipse: A ؍ab
The area of an ellipse is given by the formula
shown, where a is the distance from the center to
the graph in the horizontal direction and b is the
distance from center to graph in the vertical
direction. Find the area of the ellipse defined by
16x2 ϩ 9y2 ϭ 144.
ᮣ
727
a2 ؉ b2
B 2
The perimeter of an ellipse can be approximated by
the formula shown, where a represents the length
of the semimajor axis and b represents the length
of the semiminor axis. Find the perimeter of the
y2
x2
ellipse defined by the equation
ϩ
ϭ 1.
49
4
64. The Perimeter of an Ellipse: P ؍2
APPLICATIONS
65. Decorative fireplaces: A bricklayer intends to
build an elliptical fireplace 3 ft high and 8 ft wide,
with two glass doors that open at the middle. The
hinges to these doors are to be screwed onto a spine
that is perpendicular to the hearth and goes through
the foci of the ellipse. How far from center will the
spines be located? How tall will each spine be?
8 ft
68. Medical procedures: The medical procedure called
lithotripsy is a noninvasive medical procedure that
is used to break up kidney and bladder stones in the
body. A machine called a lithotripter uses its
three-dimensional semielliptical shape and the foci
properties of an ellipse to concentrate shock waves
generated at one focus, on a kidney stone located at
the other focus (see diagram—not drawn to scale). If
the lithotripter has a length (semimajor axis) of 16 cm
and a radius (semiminor axis) of 10 cm, how far
from the vertex should a kidney stone be located for
the best result? Round to the nearest hundredth.
3 ft
Exercise 68
Vertex
Spines
Focus
Lithotripter
66. Decorative gardens: A retired math teacher
decides to present her husband with a beautiful
elliptical garden to help celebrate their 50th
anniversary. The ellipse is to be 8 m long and 5 m
across, with decorative fountains located at the
foci. How far from the center of the ellipse should
the fountains be located (round to the nearest 100th
of a meter)? How far apart are the fountains?
67. Attracting attention to art: As part of an art
show, a gallery owner asks a student from the local
university to design a unique exhibit that will
highlight one of the more significant pieces in the
collection, an ancient sculpture. The student
decides to create an elliptical showroom with
reflective walls, with a rotating laser light on a
stand at one focus, and the sculpture placed at the
other focus on a stand of equal height. The laser
light then points continually at the sculpture as it
rotates. If the elliptical room is 24 ft long and 16 ft
wide, how far from the center of the ellipse should
the stands be located (round to the nearest 10th of a
foot)? How far apart are the stands?
Exercise 69
69. Elliptical arches: In some
situations, bridges are built
using uniform elliptical
8 ft
archways as shown in the
60 ft
figure given. Find the
equation of the ellipse forming each arch if it has a
total width of 30 ft and a maximum center height
(above level ground) of 8 ft. What is the height of a
point 9 ft to the right of the center of each arch?
70. Elliptical arches: An elliptical arch bridge is built
across a one-lane highway. The arch is 20 ft across
and has a maximum center height of 12 ft. Will a
farm truck hauling a load 10 ft wide with a clearance
height of 11 ft be able to go under the bridge
without damage? (Hint: See Exercise 69.)
cob19545_ch08_716-730.qxd
12/15/10
10:52 AM
Page 728
College Algebra G&M—
728
CHAPTER 8 Analytic Geometry and the Conic Sections
8–22
71. Plumbing: By allowing the free
flow of air, a properly vented
home enables water to run freely
throughout its plumbing system,
while helping to prevent sewage
gases from entering the home.
Find the equation of the elliptical hole cut in a roof
in order to allow a 3-in. vent pipe to exit, if the roof
4
has a slope of .
3
72. Light projection:
Standing a short
distance from a wall,
Kymani’s flashlight
projects a circle of
radius 30 cm. When
holding the flashlight at
an angle, a vertical
ellipse 50 cm long is
formed, with the focus
10 cm from the vertex
(see Worthy of Note,
page 717). Find the
equation of the circle
and ellipse, and the area of the wall that each
illuminates.
75. Planetary orbits: Except for small variations, a
planet’s orbit around the Sun is elliptical with the
Sun at one focus. The aphelion (maximum distance
from the Sun) of the planet Mars is approximately
156 million miles, while the perihelion (minimum
distance from the Sun) of Mars is about 128 million
miles. Use this information to find the lengths of the
semimajor and semiminor axes, rounded to the
nearest million. If Mars has an orbital velocity of
54,000 miles per hour (1.296 million miles per day),
how many days does it take Mars to orbit the Sun?
(Hint: Use the formula from Exercise 64.)
As a planet orbits around the Sun, it traces out an
ellipse. If the center of the ellipse were placed at (0, 0)
on a coordinate grid, the Sun would be actually offcentered (located at the focus of the ellipse). Use this
information and the graphs provided to complete
Exercises 73 through 78.
79. Area of a race track: Suppose the Toronado 500 is a
car race that is run on an elliptical track. The track is
bounded by two ellipses with equations of
4x2 ϩ 9y2 ϭ 900 and 9x2 ϩ 25y2 ϭ 900, where x
and y are in hundreds of yards. Use the formula given
in Exercise 63 to find the area of the race track.
y
Sun
x
70.5 million miles
Mercury
72 million miles
Exercise 74
y
Pluto
Sun
x
3650 million miles
3540 million miles
74. Orbit of Pluto: The
approximate orbit of the
Kuiper object formerly
known as Pluto is shown in
the figure given. Find an
equation that models this
orbit.
77. Orbital velocity of Earth: The planet Earth has a
perihelion (minimum distance from the Sun) of about
91 million mi, an aphelion (maximum distance from
the Sun) of close to 95 million mi, and completes one
orbit in about 365 days. Use this information and the
formula from Exercise 64 to find Earth’s orbital
speed around the Sun in miles per hour.
78. Orbital velocity of Jupiter: The planet Jupiter has
a perihelion of 460 million mi, an aphelion of
508 million mi, and completes one orbit in about
4329 days. Use this information and the formula
from Exercise 64 to find Jupiter’s orbital speed
around the Sun in miles per hour.
Exercise 73
73. Orbit of Mercury: The
approximate orbit of the
planet Mercury is shown
in the figure given. Find
an equation that models
this orbit.
76. Planetary orbits: The aphelion (maximum distance
from the Sun) of the planet Saturn is approximately
940 million miles, while the perihelion (minimum
distance from the Sun) of Saturn is about 840 million
miles. Use this information to find the lengths of the
semimajor and semiminor axes, rounded to the
nearest million. If Saturn has an orbital velocity of
21,650 miles per hour (about 0.52 million miles per
day), how many days does it take Saturn to orbit the
Sun? How many years?
Exercise 80
80. Area of a border: The
tablecloth for a large oval table
is elliptical in shape. It is
designed with two concentric
ellipses (one within the other)
as shown in the figure. The
equation of the outer ellipse is 9x2 ϩ 25y2 ϭ 225,
and the equation of the inner ellipse is
4x2 ϩ 16y2 ϭ 64 with x and y in feet. Use the
formula given in Exercise 63 to find the area of the
border of the tablecloth.
cob19545_ch08_716-730.qxd
12/15/10
10:52 AM
Page 729
College Algebra G&M—
8–23
Mid-Chapter Check
729
81. Whispering galleries: Due to their unique properties, ellipses are
used in the construction of whispering galleries like those in
St. Paul’s Cathedral (London) and Statuary Hall in the U.S.
Capitol. Regarding the latter, it is known that John Quincy Adams
(1767–1848), while a member of the House of Representatives,
situated his desk at a focal point of the elliptical ceiling, easily
eavesdropping on the private conversations of other House
members located near the other focal point. Suppose a whispering gallery was built using the equation
y2
x2
ϩ
ϭ 1, with the dimensions in feet. (a) How tall is the ceiling at its highest point? (b) How wide is
2809
2025
the gallery vertex to vertex? (c) How far from the base of the doors at either end, should a young couple stand so
that one can clearly hear the other whispering, “I love you.”?
82. While an elliptical billiard table has little practical value, it offers an excellent illustration of elliptical properties.
A ball placed at one focus and hit with the cue stick from any angle, will hit the cushion and immediately
rebound to the other focus and continue through each focus until coming to rest. Suppose one such table was
y2
x2
ϭ 1 as a model, with the dimensions in feet. (a) How far apart are the
constructed using the equation ϩ
9
4
vertices? (b) How far apart are the foci? As a side note, Lewis Carroll (1832–1898) did invent a game of circular
billiards, complete with rules.
ᮣ
EXTENDING THE CONCEPT
83. For 6x2 ϩ 36x ϩ 3y2 Ϫ 24y ϩ 74 ϭ Ϫ28, does
the equation appear to be that of a circle, ellipse, or
parabola? Write the equation in factored form.
What do you notice? What can you say about the
graph of this equation?
ᮣ
84. Algebraically verify that for the ellipse
y2
x2
ϩ
ϭ 1 with b 7 a, the length of the focal
a2
b2
2a2
chord is still
.
b
MAINTAINING YOUR SKILLS
85. (5.4) Evaluate the expression using the change-ofbase formula: log320.
z1
86. (3.1) Compute the product z1z2 and quotient of:
z2
z1 ϭ 213 ϩ 2i13; z2 ϭ 5 13 Ϫ 5i
87. (2.3) Solve the absolute value inequality
(a) graphically and (b) analytically:
Ϫ2Ϳx Ϫ 3Ϳ ϩ 10 7 4.
88. (2.6) The resistance R to current flow in an
electrical wire varies directly as the length L of the
wire and inversely as the square of its diameter d.
(a) Write the equation of variation; (b) find the
constant of variation if a wire 2 m long with
diameter d ϭ 0.005 m has a resistance of 240 ohms
(⍀); and (c) find the resistance in a similar wire
3 m long and 0.006 m in diameter.
MID-CHAPTER CHECK
Sketch the graph of each conic section.
1. 1x Ϫ 42 2 ϩ 1y ϩ 32 2 ϭ 9
2. x2 ϩ y2 Ϫ 10x ϩ 4y ϩ 4 ϭ 0
3.
1x Ϫ 22 2
16
ϩ
1y ϩ 32 2
1
ϭ1
4. 9x2 ϩ 4y2 ϩ 18x Ϫ 24y ϩ 9 ϭ 0
5.
1x ϩ 32 2
9
ϩ
1y Ϫ 42 2
4
ϭ1
6. 9x2 ϩ 16y2 Ϫ 36x ϩ 96y ϩ 36 ϭ 0
7. Find the equation for all points located an equal
distance from the point (0, 3) and the line
y ϭ Ϫ3.
cob19545_ch08_716-730.qxd
10/25/10
2:45 PM
Page 730
College Algebra G&M—
730
8–24
CHAPTER 8 Analytic Geometry and the Conic Sections
9. Find the equation of the ellipse having foci at
(0, 13) and 10, Ϫ132 , with a minor axis of length
10 units.
8. Find the equation of each relation and state its
domain and range.
a.
(Ϫ3, 5)
(Ϫ5, 1)
b.
y
y
10
8
6
4
(Ϫ1, 2) 2
5
4
3
2
(Ϫ1, 1) 1
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1
Ϫ1
Ϫ2
Ϫ3
(Ϫ3, Ϫ3)Ϫ4
Ϫ5
Ϫ10Ϫ8Ϫ6Ϫ4Ϫ2
Ϫ2
Ϫ4
Ϫ6
Ϫ8
Ϫ10
1 2 3 4 5 x
(3, 6)
(7, 2)
2 4 6 8 10 x
10. Find the equation of the ellipse (in standard form)
if the vertices are (Ϫ4, 0) and (4, 0) and the
distance between the foci is 4 13 units.
(3, Ϫ2)
REINFORCING BASIC CONCEPTS
More on Completing the Square
From our work so far in Chapter 8, we realize the process of completing the square has much greater use than simply as a tool
for working with quadratic equations. It is a valuable tool in the application of the conic sections, as well as other areas. The
purpose of this Reinforcing Basic Concepts is to strengthen the ability and confidence needed to apply the process correctly.
This is important because in some cases the values of a and b are rational or irrational numbers. No matter what the context,
1. The process begins with a coefficient of 1. For 20x2 ϩ 120x ϩ 27y2 Ϫ 54y ϩ 192 ϭ 0, we recognize the
equation of an ellipse, since the coefficients of the squared terms are positive and unequal. To study or graph this
ellipse, we’ll use the standard form to identify the values of a, b, and c. Grouping the like-variable terms gives
120x2 ϩ 120x
2 ϩ 127y2 Ϫ 54y
2 ϩ 192 ϭ 0
and to complete the square, we factor out the lead coefficient of each group (to get a coefficient of 1):
201x2 ϩ 6x
2 ϩ 271y2 Ϫ 2y
2 ϩ 192 ϭ 0
Subtracting 192 from both sides brings us to the fundamental step for completing the square.
2
1
2. The quantity a # linear cofficientb will complete a trinomial square. For this example we obtain
2
2
2
1
1
a # 6b ϭ 9 for x, and a # Ϫ2b ϭ 1 for y, with these numbers inserted in the appropriate group:
2
2
201x2 ϩ 6x ϩ 92 ϩ 271y2 Ϫ 2y ϩ 12 ϭ Ϫ192
complete the square
Due to the distributive property, we have in effect added 20 # 9 ϭ 180 and 27 # 1 ϭ 27 (for a total of 207) to the left
side of the equation:
201x2 ϩ 6x ϩ 92 ϩ 271y2 Ϫ 2y ϩ 12 ϭ Ϫ192
adds 20 # 9 ϭ 180
adds 27 # 1 ϭ 27
to left side
to left side
This brings us to the final step.
3. Keep the equation in balance. Since the left side was increased by 207, we also increase the right side by 207.
201x2 ϩ 6x ϩ 92 ϩ 271y2 Ϫ 2y ϩ 12 ϭ Ϫ192 ϩ 207
adds 20 # 9 ϭ 180
adds 27 # 1 ϭ 27
add 180 ϩ 27 ϭ 207
to left side
to left side
to right side
The quantities in parentheses factor, giving 201x ϩ 32 ϩ 271y Ϫ 12 ϭ 15. We then divide by 15 and simplify, obtaining
2
41x ϩ 32 2
91y Ϫ 12 2
ϭ 1. Note the coefficient of each binomial square is not 1, even after setting
3
5
the equation equal to 1. In the Strengthening Core Skills feature of this chapter, we’ll look at how to write equations of
this type in standard form to obtain the values of a and b. For now, practice completing the square using these exercises.
the standard form
ϩ
2
Exercise 1: 100x2 Ϫ 400x ϩ 18y2 Ϫ 108y ϩ 554 ϭ 0
Exercise 2: 28x2 Ϫ 56x ϩ 48y2 ϩ 192y ϩ 195 ϭ 0
cob19545_ch08_731-745.qxd
10/25/10
2:50 PM
Page 731
College Algebra G&M—
8.3
The Hyperbola
LEARNING OBJECTIVES
In Section 8.3 you will see
how we can:
A. Use the equation of a
hyperbola to graph
central and noncentral
hyperbolas
B. Distinguish between the
equations of circles,
ellipses, and hyperbolas
C. Locate the foci of a
hyperbola and use the
foci and other features to
write its equation
D. Solve applications
involving foci
EXAMPLE 1
ᮣ
As seen in Section 8.1 (see Figure 8.24), a hyperbola
is a conic section formed by a plane that cuts both
nappes of a right circular cone. A hyperbola has two Axis
symmetric parts called branches, which open in
opposite directions. Although the branches appear
to resemble parabolas, we will soon discover they
are actually a very different curve.
Figure 8.24
Hyperbola
A. The Equation of a Hyperbola
In Section 8.2, we noted that for the equation Ax2 ϩ By2 ϭ F,
if A ϭ B, the equation is that of a circle, if A B, the equation represents an ellipse. Both cases contain a sum of second-degree terms. Perhaps
driven by curiosity, we might wonder what happens if the equation has a difference of
second-degree terms. Consider the equation 9x2 Ϫ 16y2 ϭ 144. It appears the graph
will be centered at (0, 0) since no shifts are applied (h and k are both zero). Using the intercept method to graph this equation reveals an entirely new curve, called a hyperbola.
Graphing a Central Hyperbola
Graph the equation 9x2 Ϫ 16y2 ϭ 144 using intercepts and additional points
as needed.
Solution
9x2 Ϫ 16y2 ϭ 144
9102 2 Ϫ 16y2 ϭ 144
Ϫ16y2 ϭ 144
y2 ϭ Ϫ9
ᮣ
given
substitute 0 for x
simplify
divide by Ϫ16
Since y2 can never be negative, we conclude that the graph has no y-intercepts.
Substituting y ϭ 0 to find the x-intercepts gives
9x2 Ϫ 16y2 ϭ 144
9x2 Ϫ 16102 2 ϭ 144
9x2 ϭ 144
x2 ϭ 16
x ϭ 116 and x ϭ Ϫ 116
x ϭ 4 and x ϭ Ϫ4
(4, 0) and 1Ϫ4, 02
given
substitute 0 for y
simplify
divide by 9
square root property
simplify
x-intercepts
Knowing the graph has no y-intercepts, we select inputs greater than 4 and less
than Ϫ4 to help sketch the graph. Using x ϭ 5 and x ϭ Ϫ5 yields
9x2 Ϫ 16y2 ϭ 144
9152 2 Ϫ 16y2 ϭ 144
91252 Ϫ 16y2 ϭ 144
225 Ϫ 16y2 ϭ 144
Ϫ16y2 ϭ Ϫ81
81
y2 ϭ
16
9
9
yϭ
yϭϪ
4
4
y ϭ 2.25 y ϭ Ϫ2.25
15, 2.252 15, Ϫ2.252
8–25
given
substitute for x
5 ϭ 1Ϫ52 2 ϭ 25
2
simplify
subtract 225
9x2 Ϫ 16y2
91Ϫ52 2 Ϫ 16y2
91252 Ϫ 16y2
225 Ϫ 16y2
Ϫ16y2
divide by Ϫ16
square root property
decimal form
ordered pairs
9
4
y ϭ 2.25
1Ϫ5, 2.252
yϭ
ϭ 144
ϭ 144
ϭ 144
ϭ 144
ϭ Ϫ81
81
y2 ϭ
16
9
yϭϪ
4
y ϭ Ϫ2.25
1Ϫ5, Ϫ2.252
731