Quadratic Equation for Parallel Straight Lines
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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:
- $a x^2 + b y^2 + 2 h x y + 2 g x + 2 f y + c = 0$
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:
- $h^2 - a b = 0$
$\blacksquare$
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $17$.