Hello. In this lesson, I’ll derive the expression for finding out the distance between two parallel lines.

It’s quite straightforward – the distance between two parallel lines is the difference between the distances of the lines from a point.

This is what I’m talking about..

Let the equations of the lines be ax+by+c_{1}=0 and ax+by+c_{2}=0. Let P(x_{1}, y_{1}) be any point. The required distance d will be PA – PB.

(Note that the coefficients of x and y in both the lines will be correspondingly equal or proportional, because they are parallel. In case they’re not equal, you should transform the equation accordingly. I’ll discuss this during examples.)

Now, the algebraic distance PA will be \( \frac{ax_1+by_1+c_1}{\sqrt{a^2+b^2}}\). And PB = \( \frac{ax_1+by_1+c_2}{\sqrt{a^2+b^2}}\).

Therefore, **\( d = \frac{|c_1-c_2|}{\sqrt{a^2+b^2}} \)**. (The modulus takes care of the negative values)

## Lesson Summary

- The distance between the parallel lines ax+by+c
_{1}=0 and ax+by+c_{2}=0 is given by \( d = \frac{|c_1-c_2|}{\sqrt{a^2+b^2}} \)

That’s all for today. I’ll discuss a few examples related to this formula in the next lesson. Goodbye.