Quadratic Equation for Parallel Straight Lines

Theorem
Let $\LL_1$ and $\LL_2$ represent $2$ straight lines in the Cartesian plane which are represented by a quadratic equation $E$ in two variables:


 * $a x^2 + b y^2 + 2 h x y + 2 g x + 2 f y + c = 0$

Let $\LL_1$ and $\LL_2$ be parallel.

Then:
 * $h^2 - a b = 0$

Proof
From Homogeneous Quadratic Equation representing Coincident Straight Lines, $\LL_1$ and $\LL_2$ are parallel respectively to the $2$ coincident straight lines through the origin $\LL'_1$ and $\LL'_2$ represented by the homogeneous quadratic equation:


 * $a x^2 + 2 h x y + b y^2$

where:
 * $h^2 - a b = 0$

Hence $\LL_1$ and $\LL_2$ are represented by the homogeneous quadratic equation:


 * $a x^2 + b y^2 + 2 h x y + 2 g x + 2 f y + c = 0$

where:
 * $h^2 - a b = 0$