General Equation of a Line

description

• Different forms of Ax + By + C = 0 - Slope-intercept form, Intercept form, Normal form

notes

In earlier , we have  studied general equation of first degree in two variables, Ax +By+C=0 , where A,B and C are real constants such that A and B are not zero simultaneously.
Therefore, any equation of the form Ax + By + C = 0, where A and B are not zero simultaneously is called general linear equation or general equation of a line.

Different forms of Ax + By + C = 0 :
1)  Slope-intercept form :
If B ≠ 0, then Ax + By + C = 0 can be written as
y = -A/Bx-C/B
or y =mx+c           ...(1)
where m=-A/B and c = -C/B.
Equation (1) is the slope - intercept form of the equation of a line whose slope is -A/B, and y-intercept is -C/B.

2) Intercept form :
If C ≠ 0, then Ax + By + C = 0 can be written as
x/(-C/A)+y/(-C/B)= 1 or x/a+y/b=1  .....(2)
where a =-C/A and b=-C/B.
We know that equation (2) is intercept form of the equation of a line whose x-intercept is -C/A and y-intercept is -C/B.
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.

3) Normal form :
Let x cos ω + y sin ω = 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
same and  therefore, A/cosω = B/sin ω = -C/p

which gives cos ω=-(Ap)/C and sin ω = -(Bp)/C.
Now sin^2ω + cos^2ω = (-(Ap)/C)^2 + (-(Bp)/C)^2 = 1
or p^2 =C^2/(A_2+B_2) or p =+- C/sqrt(A^2+B^2)
Therefore cos ω =+- A/sqrt(A^2+B^2) and sin ω = +- B/sqrt(A^2+B^2).
Thus, the normal form of the equation Ax + By + C = 0 is      x cos ω + y sin ω = p,
where cos ω = +- A/sqrt(A^2+B^2) , sin ω = +-B/sqrt(A^2+B^2) and p = +- C/sqrt(A^2+B^2).
Proper choice of signs is made so that p should be positive.

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