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) \[2{{x}^{2}}+5xy+2{{y}^{2}}+4x+5y+2=0\]
Correct Answer: D
Solution :
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 \[2{{x}^{2}}+5xy+2{{y}^{2}}+4x+5y+2=0.\]
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