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  • The combined equation of the asymptotes of the hyperbola <img alt="\mathrm{2 x^2+5 x y+2 y^2+4 x+5 y=0}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B2%20x%5E2&plus;5%2

The combined equation of the asymptotes of the hyperbola \mathrm{2 x^2+5 x y+2 y^2+4 x+5 y=0}

Option: 1

\mathrm{2 x^2+5 x y+2 y^2=0}


Option: 2

\mathrm{2 x^2+5 x y+2 y^2-4 x+5 y+2=0=0}


Option: 3

\mathrm{2 x^2+5 x y+2 y^2+4 x+5 y-2=0}


Option: 4

\mathrm{2 x^2+5 x y+2 y^2+4 x+5 y+2=0}


Answers (1)

best_answer

Given, equation of hyperbola \mathrm{2 x^2+5 x y+2 y^2+4 x+5 y=0}  and equation of asymptotes 
\mathrm{2 x^2+5 x y+2 y^2+4 x+5 y+\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 \mathrm{a x^2+2 h x y+b y^2+2 g x+2 f y+c=0}
Comparing equation \mathrm{(i)} with standard equation, we get \mathrm{a=2, b=2, \quad h=\frac{5}{2}, g=2, f=\frac{5}{2}} and \mathrm{c=\lambda }
We also know that the condition for a pair of straight lines is \mathrm{a b c+2 f g h-a f^2-b g^2-c h^2=0}
Therefore, \mathrm{4 \lambda+25-\frac{25}{2}-8-\frac{25}{4} \lambda=0 \text { or } \frac{-9 \lambda}{4}+\frac{9}{2}=0 \text { or } \lambda=2}
Substituting value of \lambda in equation \mathrm{(i)}, we get \mathrm{2 x^2+5 x y+2 y^2+4 x+5 y+2=0}

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