Equation of straight lines :
1.General form
The first-degree equation in x and y is known as a general form of a straight line.
Example – ax + by + c = 0 , where a and b are not both zero at a same time.
x = 0 , Y-axis.
y = 0 , X-axis
x = a , parallel to Y-axis.
Y = b, parallel to X-axis.
The distance of a point (x, y ) from x-axis is |y| and from y-axis is |x|.
Slope = -a/b
Intercept on x-axis :
Put y=0.
x = -c/a
Intercept on y-axis:
Put x = 0
y = -c/b
2.Slope form
An equation of a line has slope m = tanθ and cutting off an intercept c on Y-axis.
Then
y = x.tanθ + c
y = mx+c
3. Intercept form
An equation of a straight line L and cutting of an intercepts a and b upon axes of X and Y respectively.
Then
x/a + y/b = 1
if the equation of a straight line L is –
a) parallel to X-axis, X-intercept is not defined.
b) parallel to Y-axis, Y-intercept is not defined.
4.Point-slope form
An equation of a line passing through one point and having slope m.
Then
y-y₁ = m.(x-x₁).
Here m= tanθ
5.Two-point form
An equation of a straight line passing through two points (x₁ , y₁ ) and (x₂, y₂).
Then
y – y₁ = [ (y₂ – y₁)/( x₂ -x₁ )]( x – x₁)
6.Normal form
An equation of the straight line at which the perpendicular from the origin is of length p and makes an angle a with the positive direction of the x-axis is –
Then equation of a straight line is: x cos a + y sin a = p
OM = P
7.Parametric form—
An equation of a straight line passing through A( x₁ , y₁ ) and making an angle θ with the positive direction of the x-axis.
In this equation r is the directed distance between the points A( x₁ , y₁ ) and P( x, y ).
x – x₁ = r cosθ
x = x₁ + r cosθ
Similarly ,
y – y₁ = r sinθ
y = y₁ + r sinθ
So, co-ordinate of any point on the line at a distance r from point A( x₁ , y₁ )
can be taken as ( x₁ + r cosθ , y₁ + r sinθ )
Hence r is positive for above points A and negative for points below A and 0< θ ≤ 2𝜋