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General Equation of a Line

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  • 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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