# Slope intercept form

The equation of a line with slope

*m*and making an intercept*c*on the*y*-axis is*y*=*mx*+*c*.# Point slope form

The equation of a line which passes through the point (

*x*_{1},*y*_{1}) and has the slope “*m*” is*y*–*y*_{1}=*m*(*x*–*x*_{1}).# Two point form

The equation of a line passing through two points (

*x*_{1},*y*_{1}) and (*x*_{2},*y*_{2}) is*y*–

*y*

_{1}= (

*x*–

*x*

_{1})

# Intercept form

The equation of a line which cuts off intercepts

*a*and*b*respectively from the*x*- and*y*-axes is .# Normal form

The equation of the straight line upon which the length of the perpendicular from the origin is

*p*and this perpendicular makes an angle*α*with the*x*-axis is*x*cos*α*+*y*sin*α*=*p*.# Equation of a line parallel to a given line

The equation of a line parallel to a given line

*ax*+*by*+*c*= 0 is*ax*+*by*+*λ*= 0, where*λ*is a constant.

**The value of**

*Note:**λ*can be determined by some given conditions.

# Equation of a line perpendicular to a given line

The equation of a line perpendicular to a given line

*ax*+*by*+*c*= 0 is*bx*–*ay*+*λ*= 0, where*λ*is a constant.# Equations of straight lines through (*x*_{1}, *y*_{1}) making ∠*α* with *y* = *mx* + *c*

*y*–

*y*

_{1}= tan(

*θ*±

*α*) (

*x*–

*x*

_{1}), where

*m*= tan

*θ*

# Distance form of a line (parametric form)

The equation of the straight line passing through (

*x*_{1},*y*_{1}) and making an angle*θ*with the positive direction of the*x*-axis iswhere

*r*is the distance of the point (*x*,*y*) on the line from point (*x*_{1},*y*_{1}).

*Notes:*- The equation of the line is =
*r*

⇒

*x*–*x*_{1}=*r*cos*θ*and*y*–*y*_{1}=*r*sin*θ*⇒

*x*=*x*_{1}+*r*cos*θ*and*y*=*y*_{1}=*r*sin*θ*Thus, the coordinates of any point on the line at a distance

*r*from the given point (*x*_{1},*y*_{1}) are (*x*_{1}+*r*cos*θ*,*y*_{1}+*r*sin*θ*). If*P*is on the right side of (*x*_{1},*y*_{1}), then*r*is positive and if*P*is on the left side of (*x*_{1},*y*_{1}), then*r*is negative. Since different values of*r*determine different points on the line, therefore the above form of the line is also called parametric form or symmetric form of a line.- At a given distance
*r*from the point (*x*_{1},*y*_{1}) on the line , there are two points, viz.*x*_{1}+*r*cos*θ*,*y*_{1}+*r*sin*θ*) and (*x*_{1}–*r*cos*θ*,*y*_{1}–*r*sin*θ*)