Distance between two parallel lines by having linear equations I wonder where does this formula is coming from?
It is for finding the distance between two parallel lines when we have their linear equation:
First line is:$ax+by+c=0$
Second line is:$ax+by+c_1=0$
Their distance :$$\frac{|c-c_1|}{\sqrt{a^2+b^2}}$$

Alternatively: the perpendicular line that passes through origin is: $bx-ay=0$. It crosses the two parallel lines at:
$$\left(-\frac{ac}{a^2+b^2},-\frac{bc}{a^2+b^2}\right) \ \ \text{and} \ \ \left(-\frac{ac_1}{a^2+b^2},-\frac{bc_1}{a^2+b^2}\right).$$
The distance between these points is:
$$d=\sqrt{\left(\frac{a(c-c_1)}{a^2+b^2}\right)^2+\left(\frac{b(c-c_1)}{a^2+b^2}\right)^2}=\frac{|c-c_1|}{\sqrt{a^2+b^2}}.$$
