Homogeneous Quadratic Equation for Straight Lines Parallel to those Passing through Origin

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

Then $\LL_1$ and $\LL_2$ are parallel respectively to the $2$ straight lines through the origin represented by the homogeneous quadratic equation:


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

Proof
From Characteristic of Quadratic Equation that Represents Two Straight Lines we have the conditions in which $E$ does indeed represent $2$ straight lines.

Let $E$ be written as:

Comparing coefficients of equivalent terms:

Hence:


 * $a x^2 + 2 h x y + b y^2 = b \paren {y - \mu_1 x} \paren {y - \mu_2 x}$

From Homogeneous Quadratic Equation represents Two Straight Lines through Origin, it follows that:
 * $y = \mu_1 x$
 * $y = \mu_2 x$

are two straight lines through the origin represented by the homogeneous quadratic equation:


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

By definition of slope of a straight line, these have the same slope as the straight lines $\LL_1$ and $\LL_2$:


 * $y = \mu_1 x + b_1$
 * $y = \mu_2 x + b_2$

which are described by $E$.