General Second-Degree Equation Representing a Pair of Lines

So far every pair of lines considered has passed through the origin, giving a purely homogeneous equation. In general, a pair of straight lines anywhere in the plane is represented by the general second-degree equation

but not every such equation actually splits into two straight lines — most represent a genuine conic (an ellipse, parabola, or hyperbola) instead. We need a condition on $a,h,b,g,f,c$ that decides when the equation does factor into two linear expressions.

Treat the equation as a quadratic in $x$ (assuming $a\neq0$):

For this to split into two linear factors of the form $(\ldots)(\ldots)$ — each linear in both $x$ and $y$ — the value of $x$ obtained from the quadratic formula must be a linear expression in $y$, not one involving a genuine square root of $y$. That is, the discriminant

must itself be a perfect square when viewed as a quadratic in $y$. Expanding,

and a quadratic $Ay^2+2By+C$ is a perfect square exactly when its own discriminant vanishes: $B^2-AC=0$. Here $A=h^2-ab$, $B=hg-af$, $C=g^2-ac$, so the condition is

Expanding this and simplifying (a routine but lengthy algebraic exercise) reduces it to the compact symmetric form

$abc + 2fgh - af^2 - bg^2 - ch^2 = 0,$ …