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:

ax + by + c = 0\,\,\,\,{\text{ - - - }}\left( {\text{i}} \right)


{a_1}x + {b_1}y + {c_1} = 0\,\,\,\,{\text{ - - - }}\left( {{\text{ii}}} \right)

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

d = \frac{{\left| {{a_1}{x_1} + {b_1}{y_1} + {c_1}} \right|}}{{\sqrt {{a_1}^2 + {b_1}^2} }}

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

Considers two parallel lines

\begin{gathered} ax + by + c = 0 \\ ax + by + {c_1} = 0 \\ \end{gathered}

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

d = \frac{{\left| {c - {c_1}} \right|}}{{\sqrt {{a^2} + {b^2}} }}

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.

\begin{gathered} 3x - 0 + 3 = 0 \\ \Rightarrow 3x = - 3 \Rightarrow x = - 1 \\ \end{gathered}

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

d = \frac{{\left| {3\left( { - 1} \right) - 4\left( 0 \right) + 7} \right|}}{{\sqrt {{6^2} + {{\left( { - 8} \right)}^2}} }} = \frac{{\left| { - 3 - 0 + 7} \right|}}{{\sqrt {36 + 64} }} = \frac{2}{5}