Explanation

(d)
Given equation is
${3x}^{2} + xy - y^{2} - 3x + 6y + k = 0$
$Here,\ a = 3,\ b = - 1,\ h = \frac{1}{2},\ g = - \frac{3}{2}\ f = 3,\ c = k,$
Given equation represents a pair of straight line, if
$abc + 2fgh - af^{2} - bg^{2} - {ch}^{2} = 0$
$\therefore 3( - 1)(k) + 2 \times 3 \times \left( - \frac{3}{2} \right) \times \frac{1}{2} - 3(3)^{2} + 1\left( \frac{- 3}{2} \right)^{2} - k\left( \frac{1}{2} \right)^{2} = 0$
$\Rightarrow \ - 3k - \frac{9}{2} - 27 + \frac{9}{4} - \frac{k}{4} = 0\ \Rightarrow \ k = - 9$