Question

If \( ax^2 + 2hxy + by^2 = 0 \) represents a pair of lines and \( h^2 = ab \neq 0 \), then find the ratio of their slopes.

Hint:

For a pair of lines, use the quadratic formula for slopes; \( h^2 = ab \) implies equal slopes.

Answer 1

The equation \( ax^2 + 2hxy + by^2 = 0 \) represents two straight lines through the origin. Let the slopes of the lines be \( m_1, m_2 \). The equation can be written as: \[ by^2 + 2hxy + ax^2 = 0 \quad \Rightarrow \quad b \left( \frac{y}{x} \right)^2 + 2h \left( \frac{y}{x} \right) + a = 0. \] Let \( m = \frac{y}{x} \). Then: \[ bm^2 + 2hm + a = 0. \] Roots \( m_1, m_2 \) are the slopes. Given \( h^2 = ab \), the discriminant of the quadratic in \( m \) is: \[ \Delta = (2h)^2 - 4 \cdot b \cdot a = 4h^2 - 4ab = 4(ab) - 4ab = 0. \] Since the discriminant is zero, the roots are equal: \( m_1 = m_2 \). Sum of roots: \[ m_1 + m_2 = -\frac{2h}{b}, \quad m_1 m_2 = \frac{a}{b}. \] Since \( m_1 = m_2 \), the ratio of slopes \( m_1 : m_2 = 1 : 1 \).