Homogeneous Quadratic Equation represents Two Straight Lines through Origin

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