# 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

\[d = \frac{{\left| {3\left( { – 1} \right) – 4\left( 0 \right) + 7} \right|}}{{\sqrt {{6^2} + {{\left( { – 8} \right)}^2}} }} = \frac{{\left| { – 3 – 0 + 7} \right|}}{{\sqrt {36 + 64} }} = \frac{4}{10}\]

SK

September 17@ 5:14 am-3+7 is 4 not 2.

eMathZone

September 17@ 12:25 pmThank you, we’ve now corrected this

shani

November 26@ 6:07 amI think there’s something wrong with the answer…..

christin

March 9@ 7:53 pmi think the answer is wrong you’ve considered equation two leaving out equation one ..from my calculation the answer is 4/15