# 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.