Homogeneous Second Degree Equation as a Pair of Lines

Represents a pair of straight lines through the origin if and only if h² ≥ ab. The two lines are real and distinct (h²>ab), coincident (h²=ab), or imaginary (h²<ab).

Two lines through the origin $y = m_1x$ and $y = m_2x$ have combined equation:

$$(y - m_1x)(y - m_2x) = 0$$

Expanding:

$$y^2 - (m_1+m_2)xy + m_1m_2\, x^2 = 0$$

The slopes $m_1, m_2$ are roots of the quadratic $bm^2 + 2hm + a = 0$, giving Vieta's relations:

$$m_1 + m_2 = -\frac{2h}{b}, \qquad m_1m_2 = \frac{a}{b}$$

Substituting and multiplying through by $b$:

$$\boxed{ax^2 + 2hxy + by^2 = 0}$$

Condition for real lines

Discriminant of $bm^2 + 2hm + a = 0$ is $4(h^2 - ab)$:

• $h^2 > ab$: two real distinct lines
• $h^2 = ab$: coincident lines
• $h^2 < ab$: no real lines

Angle between the pair

$$\tan\theta = \frac{2\sqrt{h^2-ab}}{|a+b|}$$

Derived by applying the angle formula to the slopes using Vieta's relations (see the angle formula derivation).

Perpendicularity

$\theta = 90° \Rightarrow a + b = 0$.

Angle bisectors

The combined equation of the bisectors of $ax^2+2hxy+by^2=0$ is:

$$\frac{x^2-y^2}{a-b} = \frac{xy}{h}$$

The bisectors are always perpendicular to each other.