The distance between the lines $y=m x+c_{1}$ and $y$ $=m x+c_{2}$ is

Let the distance between the lines be d

$\Rightarrow d=\left|\frac{y-m x-c_{1}}{\sqrt{1+m^{2}}}\right|$

Also we know that

$y-m x=c_{2}$

$\Rightarrow d=\left|\frac{c_{2}-c_{1}}{\sqrt{1+m^{2}}}\right|$

$=\left|\frac{c_{1}-c_{2}}{\sqrt{1+m^{2}}}\right|$