Page 134 - DMTH201_Basic Mathematics-1

Unit 5: Equations of Straight Lines

1 Notes
= cot
tan
1
i.e., m = or m , m = 1
2 1 2
m
1
Conversely, if m m = 1, i.e., tan tan = 1.
1 2
Then tan = cot = tan ( + 90°) or tan ( 90°) Therefore, and differ by 90°.
Thus, lines l and l are perpendicular to each other.
1 2
Hence, two non-vertical lines are perpendicular to each other if and only if their slopes are negative
reciprocals of each other,
1
i.e., m = or m , m = 1.
m 1 2
1
Let us consider the following example.

Example: Find the slope of the lines:
1. Passing through the points (3, 2) and ( 1, 4),
2. Passing through the points (3, 2) and (7, 2),

3. Passing through the points (3, 2) and (3, 4),
4. Making inclination of 60° with the positive direction of x-axis.
Solution:

1. The slope of the line through (3, 2) and ( 1, 4) is
4 ( 2) 6 3
m .
1 3 4 2
2. The slope of the line through the points (3, 2) and (7, 2) is

2 ( 2) 0
m 0.
7 3 4
3. The slope of the line through the points (3, 2) and (3, 4) is

4 ( 2) 6
m , which is not defined.
3 3 0
4. Here inclination of the line = 60°. Therefore, slope of the line is m = tan 60° = 3.

5.2.3 Angle between Two Lines

Suppose you think about more than one line in a plane, then you find that these lines are either
intersecting or parallel. Here we will discuss the angle between two lines in terms of their
slopes.
Let L and L be two non-vertical lines with slopes m and m , respectively. If and are the
1 2 1 2 1 2
inclinations of lines L and L , respectively. Then
1 2
m = tan and m = tan .
1 1 2 2

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