Distance of a Point from a Line

The distance of a point from a line means the shortest distance — the length of the perpendicular dropped from the point onto the line. For a line written in general form and a point $P(x_1, y_1)$, this has a clean formula:

The method is mechanical, but two points must be respected. First, the line must be in general form $ax + by + c = 0$ (everything on one side, zero on the other) before the coefficients $a$, $b$, $c$ are read off — a common error is to use a line still written as $y = mx + c$. Second, the numerator carries absolute-value bars, because a distance can never be negative; substitute the point, then take the modulus.

A closely related quantity is the distance between two parallel lines. Two parallel lines can always be written with identical $x$- and $y$-coefficients as $ax + by + c_1 = 0$ and $ax + by + c_2 = 0$. The distance between them is