Explanation

(a)
The equation of the asymptotes of the hyperbola
$3x^{2} + 4y^{2} + 8xy - 8x - 4y - 6 = 0$
is $\ 3x^{2} + 4y^{2} + 8xy - 8x - 4y + \lambda = 0.$
It should represent a pair of straight lines.
$\therefore abc + 2fgh - af^{2} - bg^{2} - ch^{2} = 0$
$3 \bullet 4 \bullet \lambda + 2 \bullet ( - 2)( - 4)4 - 3( - 2)^{2} - 4( - 4)^{2} - \lambda(4)^{2} = 0$
$\Rightarrow 12\lambda + 56 - 12 - 56 - 16\lambda = 0$
$\Rightarrow - 4\lambda - 12 = 0$
$\Rightarrow \ \lambda = - 3$
$\therefore\$Required equation is
$3x^{2} + 4y^{2} + 8xy - 8x - 4y - 3 = 0$