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APPENDIX B COORDINATE GEOMETRY AND LINES
APPENDIX B COORDINATE GEOMETRY AND LINES
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57–58
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59–60
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58. a bx c 2a
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■
■
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66. Prove that
Solve for x, assuming a, b, and c are negative constants.
59. ax b c
■
b .
65. Prove that ab 苷 a
Solve for x, assuming a, b, and c are positive constants.
57. abx c bc
60.
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■
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ab
b.
2
EXAMPLE 1 Describe and sketch the regions given by the following sets.
P (a,€b)
I
2
_2
_3
2
3
4
a
IV
5
x
(_3,€_2)
2
3
4
5
x
(a) x 0
(b) y=1
(c) | y |<1
(c) Recall from Appendix A that
y 1
if and only if
1 y 1
Recall from Appendix A that the distance between points a and b on a number line is
x
y
a b 苷 b a . Thus, the distance between points P x , y and P x , y on a horizontal line must be x x and the distance between P x , y and P x , y on a vertical line must be y y . (See Figure 4.)
To nd the distance P P between any two points P x , y and P x , y , we note
2
_3
P™(¤,€fi )
fi
2
By reversing the preceding process we can start with an ordered pair a, b and arrive
at the corresponding point P. Often we identify the point P with the ordered pair a, b and
refer to the point a, b. [Although the notation used for an open interval a, b is the
1
| fi-› |
›
P¡(⁄,€› )
P£(¤,€›)
| ¤-⁄|
0
FIGURE 4
⁄
¤
x
1
1
1
3
2
1
2
2
2
3
2
1
1
1
1
2
2
2
1
1
(2,€_4)
FIGURE 2
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May not be copied, scanned, or duplicated, in whole or in part.
0
(5,€0)
1
_2
_4
x
0
(b) The set of all points with y-coordinate 1 is a horizontal line one unit above the x-axis
[see Figure 3(b)].
2
_3 _2 _1 0
_1
y=1
The given region consists of those points in the plane whose y-coordinates lie between
1 and 1. Thus, the region consists of all points that lie between (but not on) the horizontal lines y 苷 1 and y 苷 1. [These lines are shown as dashed lines in Figure 3(c) to
indicate that the points on these lines don t lie in the set.]
1
1
_4
FIGURE 1
FIGURE 3
(1,€3)
3
y
y=_1
4
(_2,€2)
1
III
x
0
y
4
_3 _2 _1 O
_1
y
y=1
Just as the points on a line can be identi ed with real numbers by assigning them coordinates, as described in Appendix A, so the points in a plane can be identi ed with ordered
pairs of real numbers. We start by drawing two perpendicular coordinate lines that intersect at the origin O on each line. Usually one line is horizontal with positive direction to
the right and is called the x-axis; the other line is vertical with positive direction upward
and is called the y-axis.
Any point P in the plane can be located by a unique ordered pair of numbers as follows.
Draw lines through P perpendicular to the x- and y-axes. These lines intersect the axes in
points with coordinates a and b as shown in Figure 1. Then the point P is assigned the
ordered pair a, b. The rst number a is called the x-coordinate of P ; the second number
b is called the y-coordinate of P. We say that P is the point with coordinates a, b, and
we denote the point by the symbol Pa, b. Several points are labeled with their coordinates in Figure 2.
3
y 1}
(a) The points whose x-coordinates are 0 or positive lie on the y-axis or to the right of it
as indicated by the shaded region in Figure 3(a).
y
y
(c ) {x, y
(b) x, y y 苷 1
SOLUTION
B Coordinate Geometry and Lines
II
(a) x, y x 0
number?
(b) Is the product of two irrational numbers always an
irrational number?
b
A11
are rational numbers.
70. (a) Is the sum of two irrational numbers always an irrational
64. Use Rule 3 to prove Rule 5 of (2).
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69. Show that the sum, difference, and product of rational numbers
62. Show that if x 3 , then 4x 13 3.
63. Show that if a b, then a a.
b
Inequality with a 苷 x y and b 苷 y.]
Triangle Inequality to show that x y 5 0.05.
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same as the notation used for a point a, b, you will be able to tell from the context which
meaning is intended.]
This coordinate system is called the rectangular coordinate system or the Cartesian
coordinate system in honor of the French mathematician Ren Descartes (1596 —1650),
even though another Frenchman, Pierre Fermat (1601—1665),invented the principles of
analytic geometry at about the same time as Descartes. The plane supplied with this coordinate system is called the coordinate plane or the Cartesian plane and is denoted by ⺢ 2.
The x- and y-axes are called the coordinate axes and divide the Cartesian plane into
four quadrants, which are labeled I, II, III, and IV in Figure 1. Notice that the rst quadrant consists of those points whose x- and y-coordinates are both positive.
68. Prove that x y x y . [Hint: Use the Triangle
■
61. Suppose that x 2 0.01 and y 3 0.04. Use the
1
2
a
苷
b
[Hint: Use Equation 4.]
67. Show that if 0 a b, then a 2 b 2.
ax b
b
c
■
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2
that triangle P1P2 P3 in Figure 4 is a right triangle, and so by the Pythagorean Theorem
we have
P P 苷 s P P 1
2
1
3
2
P2 P3
2
苷 s x2 x1
苷 sx 2 x 1 2 y2 y1 2
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May not be copied, scanned, or duplicated, in whole or in part.
2
y2 y1
2
A12
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APPENDIX B COORDINATE GEOMETRY AND LINES
APPENDIX B COORDINATE GEOMETRY AND LINES
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1
Distance Formula The distance between the points P1共x 1, y1 兲 and P2 共x 2 , y2 兲 is
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A13
1
EXAMPLE 3 Find an equation of the line through 共1, ⫺7兲 with slope ⫺ 2 .
1
ⱍ P P ⱍ 苷 s共x
1
2
2
SOLUTION Using 共3兲 with m 苷 ⫺ 2 , x 1 苷 1, and y1 苷 ⫺7, we obtain an equation of the
⫺ x 1 兲2 ⫹ 共y2 ⫺ y1 兲2
line as
y ⫹ 7 苷 ⫺ 12 共x ⫺ 1兲
which we can rewrite as
EXAMPLE 2 The distance between 共1, ⫺2兲 and 共5, 3兲 is
2y ⫹ 14 苷 ⫺x ⫹ 1
s共5 ⫺ 1兲 2 ⫹ 关3 ⫺ 共⫺2兲兴 2 苷 s4 2 ⫹ 5 2 苷 s41
SOLUTION By De nition 2 the slope of the line is
We want to nd an equation of a gi ven line L; such an equation is satis ed by the coordinates of the points on L and by no other point. To nd the equation of L we use its slope,
which is a measure of the steepness of the line.
L
2
Îy=fi-›
=rise
P¡(x¡,€y¡)
y
m=5
m=2
m=1
m= 21
m=0
m=_
FIGURE 6
which simpli es to
m=_1
m=_2
m=_5
1
2
b
Thus, the slope of a line is the ratio of the change in y, ⌬y, to the change in x, ⌬x. (See
Figure 5.) The slope is therefore the rate of change of y with respect to x. The fact that the
line is straight means that the rate of change is constant.
Figure 6 shows several lines labeled with their slopes. Notice that lines with positive
slope slant upward to the right, whereas lines with negative slope slant downward to the
right. Notice also that the steepest lines are the ones for which the absolute value of the
slope is largest, and a horizontal line has slope 0.
Now let s nd an equation of the line that passes through a gi ven point P1共x 1, y1 兲 and
has slope m. A point P共x, y兲 with x 苷 x 1 lies on this line if and only if the slope of the line
through P1 and P is equal to m; that is,
y ⫺ y1
苷m
x ⫺ x1
x
y=mx+b
y ⫺ b 苷 m共x ⫺ 0兲
x
0
4
3
Point-Slope Form of the Equation of a Line An equation of the line passing through
the point P1共x 1, y1 兲 and having slope m is
y ⫺ y1 苷 m共x ⫺ x 1 兲
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Slope-Intercept Form of the Equation of a Line An equation of the line with slope m
and y-intercept b is
y 苷 mx ⫹ b
y
In particular, if a line is horizontal, its slope is m 苷 0, so its equation is y 苷 b, where
b is the y-intercept (see Figure 8). A vertical line does not have a slope, but we can write
its equation as x 苷 a, where a is the x-intercept, because the x-coordinate of every point
on the line is a.
Observe that the equation of every line can be written in the form
y=b
b
and we observe that this equation is also satis ed when x 苷 x 1 and y 苷 y1 . Therefore, it is
an equation of the given line.
This simpli es as follo ws.
FIGURE 7
This equation can be rewritten in the form
y ⫺ y1 苷 m共x ⫺ x 1 兲
3x ⫹ 2y 苷 1
Suppose a nonvertical line has slope m and y-intercept b. (See Figure 7.) This means it
intersects the y-axis at the point 共0, b兲, so the point-slope form of the equation of the line,
with x 1 苷 0 and y1 苷 b, becomes
The slope of a vertical line is not de ned.
FIGURE 5
0
y ⫺ 2 苷 ⫺ 32 共x ⫹ 1兲
⌬y
y2 ⫺ y1
苷
⌬x
x2 ⫺ x1
x
3
⫺4 ⫺ 2
苷⫺
3 ⫺ 共⫺1兲
2
Using the point-slope form with x 1 苷 ⫺1 and y1 苷 2, we obtain
Definition The slope of a nonvertical line that passes through the points
m苷
Îx=¤-⁄
=run
0
y
m苷
P1共x 1, y1 兲 and P2 共x 2 , y2 兲 is
P™(x™,€y™)
x ⫹ 2y ⫹ 13 苷 0
EXAMPLE 4 Find an equation of the line through the points 共⫺1, 2兲 and 共3, ⫺4兲.
Lines
y
or
x=a
0
a
x
5
Ax ⫹ By ⫹ C 苷 0
FIGURE 8
because a vertical line has the equation x 苷 a or x ⫺ a 苷 0 ( A 苷 1, B 苷 0, C 苷 ⫺a) and
a nonvertical line has the equation y 苷 mx ⫹ b or ⫺mx ⫹ y ⫺ b 苷 0 ( A 苷 ⫺m, B 苷 1,
C 苷 ⫺b). Conversely, if we start with a general rst-de gree equation, that is, an equation
of the form (5), where A, B, and C are constants and A and B are not both 0, then we can
show that it is the equation of a line. If B 苷 0, the equation becomes Ax ⫹ C 苷 0 or
x 苷 ⫺C兾A, which represents a vertical line with x-intercept ⫺C兾A. If B 苷 0, the equation
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A14
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APPENDIX B COORDINATE GEOMETRY AND LINES
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APPENDIX B COORDINATE GEOMETRY AND LINES
C
A
x⫺
B
B
y ⫺ 2 苷 ⫺ 23 共x ⫺ 5兲
and we recognize this as being the slope-intercept form of the equation of a line
(m 苷 ⫺A兾B, b 苷 ⫺C兾B ). Therefore, an equation of the form (5) is called a linear
equation or the general equation of a line. For brevity, we often refer to the line
Ax ⫹ By ⫹ C 苷 0 instead of the line whose equation is Ax ⫹ By ⫹ C 苷 0.
y
5y
-
3x
0
We can write this equation as 2x ⫹ 3y 苷 16.
EXAMPLE 8 Show that the lines 2x ⫹ 3y 苷 1 and 6x ⫺ 4y ⫺ 1 苷 0 are perpendicular.
SOLUTION The equations can be written as
EXAMPLE 5 Sketch the graph of the equation 3x ⫺ 5y 苷 15.
15
=
x
(5,€0)
(0,€_3)
2
||||
1–6
1
2
y⬎⫺ x⫹
0
FIGURE 10
5
x
共4, 5兲
3. 共6, ⫺2兲,
5
2
Compare this inequality with the equation y 苷 ⫺ 12 x ⫹ 52 , which represents a line with
1
5
slope ⫺ 2 and y-intercept 2 . We see that the given graph consists of points whose
1
5
y-coordinates are larger than those on the line y 苷 ⫺ 2 x ⫹ 2 . Thus, the graph is the
region that lies above the line, as illustrated in Figure 10.
■
■
7–10
共⫺1, 3兲
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Parallel and Perpendicular Lines
共5, 7兲
4. 共1, ⫺6兲,
共⫺1, ⫺3兲
■
■
■
■
■
■
■
8. P共⫺1, 6兲,
Q共4, 11兲
Parallel and Perpendicular Lines
1. Two nonvertical lines are parallel if and only if they have the same slope.
2. Two lines with slopes m1 and m2 are perpendicular if and only if m1m2 苷 ⫺1;
that is, their slopes are negative reciprocals:
m2 苷 2
3
10. P共⫺1, ⫺4兲,
Q共⫺1, ⫺6兲
■
■
■
■
■
■
■
■
■
C共⫺4, 3兲 is isosceles.
12. (a) Show that the triangle with vertices A共6, ⫺7兲, B共11, ⫺3兲,
and C共2, ⫺2兲 is a right triangle using the converse of the
Pythagorean Theorem.
(b) Use slopes to show that ABC is a right triangle.
(c) Find the area of the triangle.
13. Show that the points 共⫺2, 9兲, 共4, 6兲, 共1, 0兲, and 共⫺5, 3兲 are the
vertices of a square.
15. Show that A共1, 1兲, B共7, 4兲, C共5, 10兲, and D共⫺1, 7兲 are vertices
4x ⫹ 6y ⫹ 5 苷 0.
of a parallelogram.
17–20
5
y 苷 ⫺3 x ⫺ 6
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May not be copied, scanned, or duplicated, in whole or in part.
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■
ⱍyⱍ 苷 1
■
■
■
■
■
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21. Through 共2, ⫺3兲, slope 6
2
7
24. Through 共⫺3, ⫺5兲, slope ⫺2
■
25. Through 共2, 1兲 and 共1, 6兲
26. Through 共⫺1, ⫺2兲 and 共4, 3兲
27. Slope 3, y-intercept ⫺2
2
28. Slope 5, y-intercept 4
29. x-intercept 1, y-intercept ⫺3
30. x-intercept ⫺8, y-intercept 6
31. Through 共4, 5兲, parallel to the x-axis
32. Through 共4, 5兲, parallel to the y-axis
33. Through 共1, ⫺6兲, parallel to the line x ⫹ 2y 苷 6
34. y-intercept 6, parallel to the line 2x ⫹ 3y ⫹ 4 苷 0
35. Through 共⫺1, ⫺2兲, perpendicular to the line 2x ⫹ 5y ⫹ 8 苷 0
1
■
37–42
Sketch the graph of the equation.
17. x 苷 3
■
36. Through ( 2 , ⫺ 3 ), perpendicular to the line 4x ⫺ 8y 苷 1
16. Show that A共1, 1兲, B共11, 3兲, C共10, 8兲, and D共0, 6兲 are vertices
of a rectangle.
SOLUTION The given line can be written in the form
■
23. Through 共1, 7兲, slope 3
Q共6, 0兲
are collinear (lie on the same line) by showing that
ⱍ AB ⱍ ⫹ ⱍ BC ⱍ 苷 ⱍ AC ⱍ.
(b) Use slopes to show that A, B, and C are collinear.
EXAMPLE 7 Find an equation of the line through the point 共5, 2兲 that is parallel to the line
20.
■
22. Through 共⫺1, 4兲, slope ⫺3
14. (a) Show that the points A共⫺1, 3兲, B共3, 11兲, and C共5, 15兲
1
m2 苷 ⫺
m1
■
Q共4, ⫺3兲
11. Show that the triangle with vertices A共0, 2兲, B共⫺3, ⫺1兲, and
Slopes can be used to show that lines are parallel or perpendicular. The following facts are
proved, for instance, in Precalculus: Mathematics for Calculus, Fourth Edition by Stewart,
Redlin, and Watson (Brooks兾Cole Publishing Co., Paci c Gro ve, CA, 2002).
2
and
Find an equation of the line that satis es the gi ven
conditions.
共b, a兲
■
■
21–36
Find the slope of the line through P and Q.
7. P共1, 5兲,
■
■
■
2. 共1, ⫺3兲,
6. 共a, b兲,
共4, ⫺7兲
■
9. P共⫺3, 3兲,
6
2
19. xy 苷 0
Find the distance between the points.
||||
5. 共2, 5兲,
2
1
B Exercises
1. 共1, 1兲,
2y ⬎ ⫺x ⫹ 5
_€€€€1
2 €x+ 5
3
Since m1m2 苷 ⫺1, the lines are perpendicular.
x ⫹ 2y ⬎ 5
y=
y 苷 2x ⫺ 4
m1 苷 ⫺ 3
ⱍ
2.5
and
from which we see that the slopes are
SOLUTION We are asked to sketch the graph of the set 兵共x, y兲 x ⫹ 2y ⬎ 5其 and we do so
by solving the inequality for y :
y
1
y 苷 ⫺3 x ⫹ 3
SOLUTION Since the equation is linear, its graph is a line. To draw the graph, we can simply nd tw o points on the line. It s easiest to nd the intercepts. Substituting y 苷 0
(the equation of the x-axis) in the given equation, we get 3x 苷 15, so x 苷 5 is the
x-intercept. Substituting x 苷 0 in the equation, we see that the y-intercept is ⫺3. This
allows us to sketch the graph as in Figure 9.
EXAMPLE 6 Graph the inequality x ⫹ 2y ⬎ 5.
FIGURE 9
A15
which is in slope-intercept form with m 苷 ⫺ 23 . Parallel lines have the same slope, so the
2
required line has slope ⫺ 3 and its equation in point-slope form is
can be rewritten by solving for y:
y苷⫺
❙❙❙❙
18. y 苷 ⫺2
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■
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2
■
■
■
■
■
■
■
Find the slope and y-intercept of the line and draw
its graph.
37. x ⫹ 3y 苷 0
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May not be copied, scanned, or duplicated, in whole or in part.
38. 2x ⫺ 5y 苷 0
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