Homogeneous Quadratic Equation represents Two Straight Lines through Origin
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Theorem
Let $E$ be a homogeneous quadratic equation in two variables:
- $E: \quad a x^2 + 2 h x y + b y^2 = 0$
Then $E$ represents $2$ straight lines in the Cartesian plane:
- $y = \dfrac {h \pm \sqrt {h^2 - a b} } b x$
Proof
From Characteristic of Quadratic Equation that Represents Two Straight Lines, $E$ represents $2$ straight lines in the Cartesian plane if and only if
- $a b c + 2 f g h - a f^2 - b g^2 - c h^2 = 0$
where in this case $c = f = g = 0$, giving:
- $a b \times 0 + 2 \times 0 \times 0 \times h - a \times 0^2 - b \times 0^2 - 0 \times h^2 = 0$
The result follows from using the Quadratic Formula on $E$.
Setting $x = 0$ gives $y = 0$, confirming that $\tuple {0, 0}$ is a point on both straight lines.
$\blacksquare$
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $15$. The homogeneous equation of second degree