Condition for Lines to be Conjugate

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

Let $\PP$ and $\QQ$ be conjugate lines with respect to $\CC$:

Then:


 * $l_1 l_2 + m_1 m_2 = \dfrac {n_1 n_2} {r^2}$

Proof
By definition of conjugate lines, $\PP$ and $\QQ$ are the polars of points $P$ and $Q$ respectively, such that $P$ lies on $\QQ$ and $Q$ lies on $\PP$.

From Coordinates of Pole of Given Polar, $P$ is given by:
 * $P = \tuple {-\dfrac {l_1} {n_1} r^2, -\dfrac {m_1} {n_1} r^2}$

We have that $P$ lies on $\QQ$.

Substituting $x = -\dfrac {l_1} {n_1} r^2$ and $y = -\dfrac {m_1} {n_1} r^2$ in the equation of $\QQ$, we obtain:

from which the result follows.