Condition for Parallel Lines

The general equation represents two parallel lines iff h² = ab (same slope condition) and af² = bg² (consistency condition). When these hold, ab − h² = 0 so the point of intersection formula breaks down — the lines do not meet.

For the general equation $ax^2+2hxy+by^2+2gx+2fy+c=0$ to represent two parallel lines:

Condition 1 — Equal slopes: Both lines have the same slope. The slope of the pair depends on $a$, $h$, $b$. Two lines are parallel (non-intersecting) iff $ab-h^2 = 0$, i.e. $h^2 = ab$.

When $h^2 = ab$: $h = \sqrt{ab}$ (taking positive root), and the equation factors as:

$$(\sqrt{a}x+\sqrt{b}y)^2 + 2gx+2fy+c = 0$$

For this to factor into two distinct linear factors (not a perfect square), we need additional conditions.

Condition 2 — Consistency: If $h^2 = ab$, the equation becomes:

$$(\sqrt{a}x+\sqrt{b}y+p)(\sqrt{a}x+\sqrt{b}y+q) = 0$$

for some $p, q$. Expanding: $a+b$ term is $(\sqrt{a}x+\sqrt{b}y)^2$, $2g = \sqrt{a}(p+q)$, $2f = \sqrt{b}(p+q)$, $c = pq$.

From $2g/2f = \sqrt{a}/\sqrt{b}$: $g\sqrt{b} = f\sqrt{a}$, i.e. $g^2b = f^2a$, or:

$$af^2 = bg^2$$

Summary: Two parallel lines iff $h^2 = ab$ AND $af^2 = bg^2$ (equivalently, $bg^2 = af^2$).

Distinct parallel lines: $\Delta = 0$ and $h^2 = ab$ and $p \neq q$. Coincident lines: additionally $p = q$, so $c = p^2$ and $g^2 = ac$, $f^2 = bc$.