Perpendicular Distance from Straight Line in Plane to Origin

Theorem

Let $L$ be the straight line embedded in the cartesian plane whose equation is given as:

$a x + b y = c$

Then the perpendicular distance $d$ between $L$ and $\tuple {0, 0}$ is given by:

$d = \size {\dfrac c {\sqrt {a^2 + b^2} } }$

Proof

From Perpendicular Distance from Straight Line in Plane to Point, the perpendicular distance $d$ between $L$ and the point $\tuple {x_0, y_0}$ is gven by:

$d = \dfrac {\size {a x_0 + b y_0 + c} } {\sqrt {a^2 + b^2} }$

The result follows by setting $x_0 = 0$ and $y_0 = 0$.

$\blacksquare$

Examples

Example $1$

Let $\LL$ be the straight line defined by the equation:

$a x - b y = 1$

The distance $d$ from $\LL$ to the origin $\tuple {0, 0}$ is given by:

$d = \dfrac 1 {\sqrt {a^2 + b^2} }$