Step-by-step Solution:

Comparing the given equation with \[ ax^2 + 2hxy + by^2 = 0 \] we get: \[ a = 1, \quad h = -c, \quad b = -7 \] Let \( m_1 \) and \( m_2 \) be the slopes of the lines represented by \[ x^2 - 2cxy - 7y^2 = 0 \] Using the sum and product of roots formulas: \[ m_1 + m_2 = \frac{-2h}{b} = \frac{2c}{-7} = -\frac{2c}{7} \quad \cdots (1) \] \[ m_1 m_2 = \frac{a}{b} = \frac{1}{-7} = -\frac{1}{7} \quad \cdots (2) \] Given that \[ m_1 + m_2 = 4m_1 m_2 \] Substituting the values: \[ -\frac{2c}{7} = 4 \left( -\frac{1}{7} \right) \] \[ -\frac{2c}{7} = -\frac{4}{7} \] \[ 2c = 4 \] \[ c = 2 \]