$$\begin{gathered} \mathrm{If} 2{\mathrm{x}}^2+5\mathrm{xy}+2{\mathrm{y}}^2-11\mathrm{x}-7\mathrm{y}+k=0 \mathrm{is} \mathrm{the} \mathrm{pair} \mathrm{of} \mathrm{asymptotes} \mathrm{of} \mathrm{the} \mathrm{hyperbola} 2{\mathrm{x}}^2+5\mathrm{xy}+2{\mathrm{y}}^2-11\mathrm{x}-7\mathrm{y}-4=0 \\ \mathrm{then} k= \end{gathered}$$

$$\begin{gathered} Given hyperbola S\equiv 2x^2+5xy+2y^2-11x-7y-4=0 \\ Now \frac{\partial S}{\partial x}=0\Rightarrow 4x+5y-11=0\to \left(1\right) \\ and \frac{\partial S}{\partial y}=0\Rightarrow 5x+4y-7=0\to \left(2\right) \\ Now point of intersection of \left(1\right) and \left(2\right) is (-1,3) \\ and given pair of asymptotes is 2x^2+5xy+2y^2-11x-7y+k=0 \\ It passes through the point (-1,3) then \\ 2\left(1\right)+5(-3)+2\left(9\right)+11-21+k=0 \\ \Rightarrow k=5 \end{gathered}$$