The distance between two parallel lines in the plane is the minimum distance between any two points.
Because the lines are parallel, the perpendicular distance between them is a constant, so it does not matter which point is chosen to measure the distance. Given the equations of two non-vertical parallel lines
y=mx+b1
y=mx+b2,
the distance between the two lines is the distance between the two intersection points of these lines with the perpendicular line
y=-x/m.
This distance can be found by first solving the linear systems
\begin{cases} y=mx+b1\\ y=-x/m, \end{cases}
and
\begin{cases} y=mx+b2\\ y=-x/m, \end{cases}
to get the coordinates of the intersection points. The solutions to the linear systems are the points
\left(x1,y1\right) =\left(
| -b1m | , | |
| m2+1 |
| b1 | |
| m2+1 |
\right),
and
\left(x2,y2\right) =\left(
| -b2m | , | |
| m2+1 |
| b2 | |
| m2+1 |
\right).
The distance between the points is
d=\sqrt{\left(
| b1m-b2m | |
| m2+1 |
\right)2+\left(
| b2-b1 | |
| m2+1 |
\right)2},
which reduces to
d=
| |b2-b1| | |
| \sqrt{m2+1 |
When the lines are given by
ax+by+c1=0
ax+by+c2=0,
the distance between them can be expressed as
d=
| |c2-c1| | |
| \sqrt{a2+b2 |