# Find the asymptotes of the hyperbola $$2x^2 + 5xy + 2y^2 + 4x + 5y$$ = 0. Find also the general equation of all the hyperbolas having the same set of asymptotes.

## Solution :

Let $$2x^2 + 5xy + 2y^2 + 4x + 5y + k$$ = 0 be asymptotes. This will represent two straight line

so $$abc + 2fgh – af^2 – bg^2 – ch^2$$ = 0 $$\implies$$ 4k + 25 – $$25\over 2$$ – 8 – $$25\over 4$$k = 0

$$\implies$$ k = 2

$$\implies$$ $$2x^2 + 5xy + 2y^2 + 4x + 5y + 2$$ = 0 are asymptotes

$$\implies$$ (2x+y+2) = 0 and (x+2y+1) = 0 are asymptotes

and  $$2x^2 + 5xy + 2y^2 + 4x + 5y + c$$ = 0 is general equation of hyperbola.

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