Nature of the Lines

The discriminant of the quadratic in m (from ax² + 2hxy + by² = 0) is 4(h²−ab). The three cases correspond to positive, zero, and negative discriminant.

The quadratic $bm^2+2hm+a=0$ has discriminant $\Delta' = (2h)^2 - 4ab = 4(h^2-ab)$.

Real and distinct lines: The equation represents two genuinely separate lines through the origin. This is the most common case in problems.

Coincident lines: $ax^2+2hxy+by^2 = (\sqrt{a}x+\sqrt{b}y)^2$ when $h = \sqrt{ab}$. Both "lines" are the same — the equation represents a double line.

Imaginary lines: The equation $ax^2+2hxy+by^2=0$ has only the trivial solution $(0,0)$ in the reals. This occurs, for example, with $x^2+xy+y^2=0$ (discriminant $= 1-4 < 0$).

Note: The condition $h^2 = ab$ for coincident lines here is the same as for parallel lines in the general equation — coincident lines are a special case of parallel lines (zero distance apart).