Distance Between Two Parallel Lines
If two lines are parallel, then they have the same distance between them throughout,
Therefore the distance between two parallel linesย \(ax + by + c_1\) and \(ax + by + c_2\) is given by :
D = \(|c_1 โ c_2|\over \sqrt{a^2 + b^2}\)
Note โ Both equation must be in the given form ย \(ax + by + c_1\) and \(ax + by + c_2\), if it is not in the given form reduce them to the given form as shown in the example below.
Example : Find the the distance between two parallel lines 3x โ 4y + 9 and 6x โ 8y โ 15 = 0.
Solution : Given lines are 3x โ 4y + 9 and 6x โ 8y โ 15 = 0.
Divide line 6x โ 8y โ 15 = 0 by 2
we get, 3x โ 4y โ 15/2 = 0.
Now both the equation are reduced to given form.
Hence, we can find the distance using above formula
D = \(|c_1 โ c_2|\over \sqrt{a^2 + b^2}\)
Required distance D = \(|9 โ (-15/2)|\over \sqrt{3^2 + (-4)^2}\)
D = \(9 + {15\over 2}\over 5\) = \(33\over 10\)
Example : Find the equation of lines parallel to 3x โ 4y โ 5 = 0 at a unit distance from it.
Solution : Equation of any line parallel to 3x โ 4y โ 5 = 0 is
3x โ 4y + \(\lambda\) = 0 โฆ..(i)
It is given that the distance between the line 3x โ 4y โ 5 = 0 and line (i) is 1 unit.
\(\therefore\) \(|\lambda โ (-5)|\over \sqrt{3^2 + (-4)^2}\) = 1
\(\implies\) \(|\lambda + 5|\over 5\) = 1
\(|\lambda + 5|\) = 5 \(\implies\) \(\lambda + 5\) = \(\pm 5\)
\(\implies\) \(\lambda\) = 0 , -10
Substituting the values of \(\lambda\) in (i), we get
3x โ 4y = 0 and 3x โ 4y โ 10 = 0
as the equations of required lines.
