If the sum of the slopes of the lines given by $x^2 - 2cxy - 7y^2 = 0$ is four times their product, then the value of $c$ is

Given: $x^2 - 2cxy - 7y^2 = 0$
Comparing with $ax^2 + 2hxy + by^2 = 0$:
$a = 1,\ h = -c,\ b = -7$
By direct formula:
$m_1 + m_2 = \dfrac{-2h}{b} = \dfrac{-2(-c)}{-7} = \dfrac{-2c}{7}$
$m_1 \cdot m_2 = \dfrac{a}{b} = \dfrac{1}{-7} = \dfrac{-1}{7}$
Given condition: $m_1 + m_2 = 4(m_1 \cdot m_2)$
$\Rightarrow \dfrac{-2c}{7} = 4 \times \dfrac{-1}{7}$
$\Rightarrow -2c = -4$
$\Rightarrow c = 2$
$\therefore \boxed{c = 2}$