(d) Given, equation of hyperbola $2{x^2} + 5xy + 2{y^2} + 4x + 5y = 0$

and equation of asymptotes $2{x^2} + 5xy + 2{y^2} + 4x + 5y + \lambda = 0$ .….$(i),$

which is the equation of a pair of straight lines.

We know that the standard equation of a pair of straight lines is $a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0.$

Comparing equation $(i)$ with standard equation,

we get $a = 2,\,b = 2,$ $\,h = \frac{5}{2},\,g = 2,\,f = \frac{5}{2}$ and $c = \lambda .$

We also know that the condition for a pair of straight lines is $abc + 2fgh – a{f^2} – b{g^2} – c{h^2} = 0.$

Therefore $4\lambda + 25 – \frac{{25}}{2} – 8 – \frac{{25}}{4}\lambda = 0$

or $ – \frac{{9\lambda }}{4} + \frac{9}{2} = 0$ or $\lambda = 2$.

Substituting value of $\lambda $ in equation $(i),$

we get $2{x^2} + 5xy + 2{y^2} + 4x + 5y + 2 = 0.$