• # question_answer The combined equation of the asymptotes of the hyperbola $2{{x}^{2}}+5xy+2{{y}^{2}}+4x+5y=0$                [Karnataka CET 2002] A)            $2{{x}^{2}}+5xy+2{{y}^{2}}=0$ B)            $2{{x}^{2}}+5xy+2{{y}^{2}}-4x+5y+2=0$ C)            $2{{x}^{2}}+5xy+2{{y}^{2}}+4x+5y-2=0$ D)            $2{{x}^{2}}+5xy+2{{y}^{2}}+4x+5y+2=0$

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