Distance between two parallel lines

Distance between two lines is the distance from a point of the first line to the another line.

Formula and proof

Because a parallel line is a line that has an equal distance with the opposite line, there is a unique distance between the two parallel lines. Given the equations of two non-vertical parallel lines

$y=mx+b_{1}\,$

$y=mx+b_{2}\,,$

the distance between the two lines can be found by solving the linear systems

${\begin{cases}y=mx+b_{1}\\y=-x/m\end{cases}}$

and

${\begin{cases}y=mx+b_{2}\\y=-x/m\end{cases}}$

to get the coordinates of the points. The solutions to the linear systems are the points

$\left(x_{1},y_{1}\right)\ =\left({\frac {-b_{1}m}{m^{2}+1}},{\frac {b_{1}}{m^{2}+1}}\right)\,$

and

$\left(x_{2},y_{2}\right)\ =\left({\frac {-b_{2}m}{m^{2}+1}},{\frac {b_{2}}{m^{2}+1}}\right).\,$

The distance between the points is

$d={\sqrt {\left({\frac {b_{1}m-b_{2}m}{m^{2}+1}}\right)^{2}+\left({\frac {b_{2}-b_{1}}{m^{2}+1}}\right)^{2}}}\,,$

which reduces to

$d={\frac {|b_{2}-b_{1}|}{\sqrt {m^{2}+1}}}\,.$

When the lines are given by

$ax+by+c_{1}=0\,$

$ax+by+c_{2}=0,\,$

their distance can be expressed as

$d={\frac {|c_{2}-c_{1}|}{\sqrt {a^{2}+b^{2}}}}.$