James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

A12
appendix B Coordinate Geometry and Lines
1 Distance Formula The distance between the points P 1 sx 1 , y 1 d and P 2 sx 2 , y 2 d is
| P 1P 2 | − ssx 2 2 x 1 d 2 1 sy 2 2 y 1 d 2
Example 2 The distance between s1, 22d and s5, 3d is
ss5 2 1d 2 1 f3 2 s22dg 2 − s4 2 1 5 2 − s41
n
Lines
We want to find an equation of a given line L; such an equation is satisfied by the coordinates
of the points on L and by no other point. To find the equation of L we use its slope,
which is a measure of the steepness of the line.
y
P¡(x¡, y¡)
0
FIGURE 5
y
P(x, y)
Îx=¤-⁄
=run
L
Îy=fi-›
=rise
m=5
m=2
m=1
m= 1 2
m=0
x
2 Definition The slope of a nonvertical line that passes through the points
P 1 sx 1 , y 1 d and P 2 sx 2 , y 2 d is
The slope of a vertical line is not defined.
m − Dy
Dx − y 2 2 y 1
x 2 2 x 1
Thus the slope of a line is the ratio of the change in y, Dy, to the change in x, Dx. (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 find an equation of the line that passes through a given point P 1 sx 1 , y 1 d
and has slope m. A point Psx, yd with x ± x 1 lies on this line if and only if the slope of
the line through P 1 and P is equal to m; that is,
0
FIGURE 6
m=_ 1 2
m=_1 x
m=_2
m=_5
y 2 y 1
x 2 x 1
This equation can be rewritten in the form
− m
y 2 y 1 − msx 2 x 1 d
and we observe that this equation is also satisfied when x − x 1 and y − y 1 . Therefore it
is an equation of the given line.
3 Point-Slope Form of the Equation of a Line An equation of the line passing
through the point P 1 sx 1 , y 1 d and having slope m is
y 2 y 1 − msx 2 x 1 d
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.