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

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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:

\(\ds b \paren {y - \mu_1 x - b_1} \paren {y - \mu_2 x - b_2}\) \(=\) \(\ds 0\)
\(\ds \leadsto \ \ \) \(\ds b y^2 - b \paren {\mu_1 + \mu_2} x y - b \paren {b_1 + b_2} y + b \mu_1 \mu_2 x^2 + b \paren {b_1 \mu_2 + b_2 \mu_2} + b b_1 b_2\) \(=\) \(\ds 0\) multiplying out


Comparing coefficients of equivalent terms:

\(\ds b \mu_1 \mu_2\) \(=\) \(\ds a\)
\(\ds \leadsto \ \ \) \(\ds \mu_1 \mu_2\) \(=\) \(\ds \dfrac a b\)
\(\ds -b \paren {\mu_1 + \mu_2}\) \(=\) \(\ds 2 h\)
\(\ds \leadsto \ \ \) \(\ds \mu_1 + \mu_2\) \(=\) \(\ds \dfrac {-2 h} b\)

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 $E$.

$\blacksquare$


Sources

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