Coordinates of Pole of Given Polar

Theorem
Let $\CC$ be a circle of radius $r$ whose center is at the origin of a Cartesian plane.

Let $\LL$ be a straight line whose equation is given as:
 * $l x + m y + n = 0$

Then the pole $P$ of $\LL$ with respect to $\CC$ is:
 * $P = \tuple {-\dfrac l n r^2, -\dfrac m n r^2}$

Proof
From Equation of Circle center Origin, we have that the equation of $\CC$ is:
 * $x^2 + y^2 = r^2$

Let $P = \tuple {x_0, y_0}$.

By definition of polar:
 * $x x_0 + y y_0 = r^2$

Comparing this with the equation for $\LL$:


 * $\dfrac {x_0} l = \dfrac {y_0} m = \dfrac {r^2} {-n}$

The result follows.