Distance Between Parallel Lines

Distance between ax + by + c₁ = 0 and ax + by + c₂ = 0.

Let the two parallel lines be $ax + by + c_1 = 0$ and $ax + by + c_2 = 0$.

Pick any point on the first line. Setting $x = 0$: the point is $\left(0, -\frac{c_1}{b}\right)$.

Its distance to the second line:

$$d = \frac{\left|a \cdot 0 + b \cdot \left(-\frac{c_1}{b}\right) + c_2\right|}{\sqrt{a^2+b^2}} = \frac{|c_2 - c_1|}{\sqrt{a^2+b^2}}$$ $$\boxed{d = \frac{|c_1 - c_2|}{\sqrt{a^2 + b^2}}}$$

Note: The two lines must have identical coefficients of $x$ and $y$ (not just proportional) before applying this formula. If given $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ with $\frac{a_1}{a_2} = \frac{b_1}{b_2} = k$, first rewrite the second line as $a_1x + b_1y + \frac{c_2}{k} = 0$, then apply the formula.