Pair of Straight Lines Through the Origin

A single second-degree equation in $x$ and $y$ can, in a special case, represent not one curve but two straight lines together. The homogeneous second-degree equation

always represents a pair of straight lines passing through the origin (it is called "homogeneous" because every term has degree exactly 2, so $(0,0)$ always satisfies it, and so does every point of the form $(kx_0, ky_0)$ once $(x_0,y_0)$ satisfies it — i.e. whole lines through the origin satisfy it).

Dividing through by $y^2$ (for $y \ne 0$) and writing $t = x/y$ turns this into an ordinary quadratic in $t$:

The two roots $t_1, t_2$ correspond to the two lines $x = t_1 y$ and $x = t_2 y$. Whether these two lines are real and distinct, real and coincident, or not real at all depends on the discriminant of this quadratic:

• Real and distinct lines when $h^2 > ab$.
• Real and coincident lines (the pair degenerates to a single repeated line) when $h^2 = ab$. …