equation of straight lines

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.

intercept form of equation

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

point slope form of equation

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

normal form of equation

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.

parametric form of equation

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

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