Find the equations of the bisectors of the angles between the pair of straight lines given by $\;$ $6x^2 + 5xy - 6y^2 = 0$
Equation of given pair of lines: $\;$ $6x^2 + 5xy - 6y^2 = 0$ $\;\;\; \cdots \; (1)$
Comparing with the standard equation of pair of lines: $\;$ $ax^2 + 2hxy + by^2 = 0$ $\cdots \; (2)$ gives
$a = 6, \;\; h = \dfrac{5}{2}, \;\; b = -6$
The combined equation of the bisectors of equation $(2)$ is $\;\;$ $\dfrac{x^2 - y^2}{a - b} = \dfrac{xy}{h}$
$\therefore \;$ The combined equation of the bisectors of equation $(1)$ is
$\dfrac{x^2 - y^2}{6 - \left(-6\right)} = \dfrac{xy}{\dfrac{5}{2}}$
i.e. $\;$ $\dfrac{x^2 - y^2}{12} = \dfrac{2 xy}{5}$
i.e. $\;$ $5 \left(x^2 - y^2\right) - 24 xy = 0$