Page 148 - DMTH201_Basic Mathematics-1

Unit 5: Equations of Straight Lines

Notes
Example: Reduce the equation 3x y 8 0 into normal form. Find the values of
p and .
Solution:
Given equation is:

3x y 8 = 0 ... (1)

2
Dividing (1) by 3 (1) 2 = 2 , we get

3 1
x y = 4 or cos 30° x + sin 30° y = 4 …(2)
2 2
Comparing (2) with x cos + y sin = p, we get p = 4 and = 30°.

Example: Find the angle between the lines y 3x 5 0 and 3y x 6 0.

Solution:
Given lines are

y 3x 5 = 0 or y 3x 5 …(1)

1
and 3y x 6 = 0 or y x 2 3 …(2)
3
1
Slope of line (1) is m 3 and slope of line (2) is m .
1 2
3
The acute angle (say) between two lines is given by

m m
tan = 2 1 ... (3)
1 m m 2
1
Putting the values of m and m in (3), we get
1 2
1
3
3 1 3 1
tan =
1 2 3 3
1 3
3
which gives = 30°. Hence, angle between two lines is either 30°or 180° – 30° = 150°.

Example: Show that two lines a x + b y + c = 0 and a x + b y + c = 0, where b , b 0 are:
1 1 1 2 2 2 1 2
a a
(i) Parallel if 1 2 , and
b 1 b 2
(ii) Perpendicular if a a + b b = 0.
1 2 1 2

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