General Second-Degree Equation

Represents a pair of straight lines (not necessarily through the origin) when the discriminant condition Δ = 0 is satisfied. May also represent a conic if Δ ≠ 0.

The general second-degree equation $S \equiv ax^2+2hxy+by^2+2gx+2fy+c=0$ represents a pair of straight lines when it factors into two linear factors:

$$S = (l_1x+m_1y+n_1)(l_2x+m_2y+n_2) = 0$$

Expanding and comparing:

$$a = l_1l_2, \quad 2h = l_1m_2+l_2m_1, \quad b = m_1m_2$$ $$2g = l_1n_2+l_2n_1, \quad 2f = m_1n_2+m_2n_1, \quad c = n_1n_2$$

For this factorization to exist: The six quantities $a, h, b, g, f, c$ must be consistent — they cannot be assigned independently. The consistency condition is exactly $\Delta = 0$.

What $S = 0$ can represent when $\Delta = 0$:

• Two real and distinct lines (if $h^2 > ab$)
• Two parallel lines (if $h^2 = ab$, lines may be distinct or coincident)
• Two coincident lines (if $h^2 = ab$ and $g^2 = ac$ and $f^2 = bc$)
• Two imaginary lines (if $h^2 < ab$, but then $\Delta = 0$ is harder to satisfy simultaneously)

When $\Delta \neq 0$: The equation represents a proper conic — ellipse, hyperbola, parabola, or circle — depending on $h^2 - ab$: