Explanation

(d)
Given equation is $2x^{2} + 5xy + 2y^{2} + 4x + 5y = 0$ and equation of its asymptotes is
$2x^{2} + 5xy + 2y^{2} + 4x + 5y + \lambda = 0$ …(i)
Which is the equation of pair of straight lines Eq.(i) is compared by the standard equation of pair of straight lines.
$\Rightarrow a = 2,\ b = 2,\ h = \frac{5}{2},\ g = 2,\ f = \frac{5}{2}$ and $c = \lambda$
The condition for a pair of straight lines is
$abc + 2fgh - af^{2} - bg^{2} - ch^{2} = 0$
$\therefore 2(2)(\lambda) + 2\left( \frac{5}{2} \right)2\left( \frac{5}{2} \right) - 2\left( \frac{5}{2} \right)^{2} - 2(2)^{2} - \lambda\left( \frac{5}{2} \right)^{2} = 0$
$\Rightarrow \ 4\lambda + 25 - \frac{25}{2} - 8 - \frac{25\lambda}{4} = 0$
$\Rightarrow \frac{25\lambda}{4} - 4\lambda = \frac{25}{2} - 8\ \Rightarrow \ \lambda = 2$
On putting the value of $\lambda$ in Eq.(i), we get
$2x^{2} + 5xy + 2y^{2} + 4x + 5y + 2 = 0$
Which is the required equation.