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

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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.