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:

When the equation represents a pair of lines, use the quadratic formula to find the slopes, and use the condition \( h^2 = ab \) to determine the relationship between the slopes.

Answer 1

The equation \( ax^2 + 2hxy + by^2 = 0 \) represents a pair of straight lines. The general form for the slopes of the lines is given by solving the quadratic equation in terms of \( x/y \) using the method of finding the roots of the quadratic equation. The roots of the quadratic equation \( ax^2 + 2hxy + by^2 = 0 \) represent the slopes of the two lines. The equation can be rewritten as: \[ a m^2 + 2h m + b = 0 \] where \( m \) is the slope of the lines. Solving for \( m \) using the quadratic formula gives the slopes of the lines. The quadratic formula for the roots of the equation \( am^2 + bm + c = 0 \) is: \[ m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] For our case, \( a = a \), \( b = 2h \), and \( c = b \). The discriminant is: \[ % Option (2h)^2 - 4ab = 4h^2 - 4ab = 0 \] So, the slopes are: \[ m_1 = m_2 = \frac{-2h}{2a} = \frac{-h}{a} \] Thus, the ratio of the slopes is: \[ \boxed{1} \]