Homogeneous Quadratic Equation representing Imaginary Straight Lines

Theorem

Let $E$ be a homogeneous quadratic equation in two variables:

$E: \quad a x^2 + 2 h x y + b y^2$

Let $h^2 - a b < 0$.

Then

Then $E$ represents a conjugate pair of imaginary straight lines in the Cartesian plane:

Proof

From Homogeneous Quadratic Equation represents Two Straight Lines through Origin, $E$ represents $2$ straight lines in the Cartesian plane:

$y = \dfrac {h \pm \sqrt {h^2 - a b} } b x$

But when $h^2 - a b$, $\sqrt {h^2 - a b}$ is not defined as a real number.

$y = \dfrac {h \pm i \sqrt {h^2 - a b} } b x$
$\blacksquare$