# Distance Between Two Parallel Lines

In order to find the distance between two parallel lines, first we find a point on one of the lines and then we find its distance from the other line.

For example, the equations of two parallel lines are:

Let $\left( {{x_1},{y_1}} \right)$ be a point on line (i); then its distance from line (ii) will be the distance between lines (i) and (ii).

Now the distance of the point $\left( {{x_1},{y_1}} \right)$ from the line (ii) is given by

Alternatively we can find the distance between two parallel lines as follows:

Considers two parallel lines

Now the distance between two parallel lines can be found with the following formula:

Example: Find the distance between the parallel lines
$3x - 4y + 3 = 0\,\,\,{\text{ - - - }}\left( {\text{i}} \right)$ and $6x - 8y + 7 = 0\,\,\,{\text{ - - - }}\left( {{\text{ii}}} \right)$

First we find a point $A$ on (i). For this, we put $y = 0$ in equation (i), i.e.

Thus, $A\left( { - 1,0} \right)$ is a point on line (i). If $d$ is the distance between the given lines (i) and (ii), then $d$ is the distance of the point $A$ from the line (ii), so