Condition for Pairs of Lines through Origin to be Harmonic Conjugates/Homogeneous Quadratic Equation Form

Theorem
Consider the $2$ Homogeneous quadratic equations:

each representing $2$ straight lines through the origin.

Then the $2$ straight lines represented by $(\text E 1)$ are harmonic conjugates of the $2$ straight lines represented by $(\text E 2)$ :


 * $a_1 b_2 + a_2 b_1 - 2 h_1 h_2 = 0$

Proof
From Homogeneous Quadratic Equation represents Two Straight Lines through Origin, $(\text E 1)$ and $(\text E 2)$ represent straight lines through the origin :

Let the $2$ straight lines represented by $(\text E 1)$ be defined by the equations:

Let the $2$ straight lines represented by $(\text E 1)$ be defined by the equations:

Then we can write the Condition for Pairs of Lines through Origin to be Harmonic Conjugates as:


 * $(1): \quad 2 \paren {\lambda_1 \mu_1 + \lambda_2 \mu_2} = \paren {\lambda_1 + \mu_1} \paren {\lambda_2 + \mu_2}$

We can express $a_1 x^2 + 2 h_1 x y + b_1 y^2 = 0$ as:
 * $b_1 \paren {y - \lambda_1 x} \paren {y - \mu_1 x} = 0$

and $a_2 x^2 + 2 h_2 x y + b_2 y^2 = 0$ as:
 * $b_2 \paren {y - \lambda_2 x} \paren {y - \mu_2 x} = 0$

from which we obtain via Sum of Roots of Quadratic Equation and Product of Roots of Quadratic Equation:

Therefore $(1)$ reduces to: