Page 147 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes If C = 0, then Ax + By + C = 0 can be written as Ax + By = 0, which is a line passing through the
origin and, therefore, has zero intercepts on the axes.

Normal Form

Let x cos + y sin = p be the normal form of the line represented by the equation Ax + By + C
= 0 or Ax + By = – C. Thus, both the equations are:
A B C
same and therefore, =
cos sin P

Ap Bp
which gives cos = and sin .
C C
Ap 2 Bp 2
2
2
Now sin + cos = 1
C C
C 2 C
2
or p = or p
A 2 B 2 A 2 B 2
A B
Therefore, cos = and sin .
A 2 B 2 A 2 B 2
Thus, the normal form of the equation Ax + By + C = 0 is

x cos + y sin = p,

A B C
where cos , sin and p .
A 2 B 2 A 2 B 2 A 2 B 2
Proper choice of signs is made so that p should be positive.

Example: Equation of a line is 3x 4y + 10 = 0. Find its (i) slope, (ii) x - and y-intercepts.
Solution:
(i) Given equation 3x 4y + 10 = 0 can be written as

3 5
y = x ... (1)
4 2
3
Comparing (1) with y = mx + c, we have slope of the given line as m .
4
(ii) Equation 3x 4y + 10 = 0 can be written as

x y
3x 4y 10 or = 1 …(2)
10 5
3 2

x y 10 5
Comparing (2) with 1, we have x-intercept as a and y-intercept as b .
a b 3 2

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