Sum and Product of Slopes

The two lines y = m₁x and y = m₂x satisfying ax² + 2hxy + by² = 0. Derived by dividing through by x² to get a quadratic in y/x = m. Valid when b ≠ 0.

Dividing $ax^2+2hxy+by^2=0$ by $x^2$ gives the quadratic in $m = y/x$:

$$bm^2 + 2hm + a = 0$$

By Vieta's formulas:

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

Usage: These relations allow solving problems without finding $m_1$ and $m_2$ individually.

Example: Find $m_1^2 + m_2^2$:

$$m_1^2+m_2^2 = (m_1+m_2)^2 - 2m_1m_2 = \frac{4h^2}{b^2} - \frac{2a}{b} = \frac{4h^2-2ab}{b^2}$$

When $b = 0$: One line is $x = 0$ (vertical, slope undefined). The other is $ax + 2hy = 0$, slope $= -a/2h$. The Vieta formulas break down in this case — treat it separately.

Joint equation from slopes: Given two lines with slopes $m_1$ and $m_2$ through the origin, the joint equation is:

$$y^2 - (m_1+m_2)xy + m_1m_2x^2 = 0$$

Substitute the known values of $m_1+m_2$ and $m_1m_2$ directly.