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 then we find its distance from the other line.
For example, consider the equations of parallel lines are given as

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 the line (i), then its distance from the line (ii) will be the distance between the 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 distance between two parallel lines as
Considers two parallel lines

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


Now distance between two parallel lines by 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)
Find 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 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}

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