How to Find the Distance Between Two Parallel Lines

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 lines  \(ax + by + c_1\) and \(ax + by + c_2\) is given by :

D = \(|c_1 – c_2|\over \sqrt{a^2 + b^2}\)

Note – Both equation must be in the given form  \(ax + by + c_1\) 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 – 4y + 9 and 6x – 8y – 15 = 0.

Solution : Given lines are 3x – 4y + 9 and 6x – 8y – 15 = 0.

Divide line 6x – 8y – 15 = 0 by 2

we get, 3x – 4y – 15/2 = 0.

Now both the equation are reduced to given form.

Hence, we can find the distance using above formula

D = \(|c_1 – c_2|\over \sqrt{a^2 + b^2}\)

Required distance D = \(|9 – (-15/2)|\over \sqrt{3^2 + (-4)^2}\)

D = \(9 + {15\over 2}\over 5\) = \(33\over 10\)

Example : Find the equation of lines parallel to 3x – 4y – 5 = 0 at a unit distance from it.

Solution : Equation of any line parallel to 3x – 4y – 5 = 0 is

3x – 4y + \(\lambda\) = 0 …..(i)

It is given that the distance between the line 3x – 4y – 5 = 0 and line (i) is 1 unit.

\(\therefore\) \(|\lambda – (-5)|\over \sqrt{3^2 + (-4)^2}\) = 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 – 4y = 0 and 3x – 4y – 10 = 0

as the equations of required lines.

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