Find the asymptotes of the curve $2 x^{2} + 5 x y + 2 y^{2} + 4 x + 5 y = 0$ , and find the general equation of all hyperbolas having the same asymptotes.

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Solution

Equation of asymyptotes and and the curve only differ by constant. So the combined equation of asmyptotes is

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

It represents a pair of straight lines

$∴ a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2} = 0 \Rightarrow 2 (2) c + 2 (\frac{5}{2}) (2) (\frac{5}{2}) - 2 {(\frac{5}{2})}^{2} - 2 {(2)}^{2} - c {(\frac{5}{2})}^{2} = 0 \Rightarrow 4 c + 25 - \frac{25}{2} - 8 - \frac{25 c}{4} = 0 \Rightarrow 16 c + 100 - 50 - 32 - 25 c = 0 \Rightarrow c = 2$

So, the equation of asymtotes is

$2 x^{2} + 5 x y + 2 y^{2} + 4 x + 5 y + 2 = 0 2 y^{2} + (5 x + 5) y + 2 x^{2} + 4 x + 2 = 0 y = \frac{- (5 x + 5) \pm \sqrt{{(5 x + 5)}^{2} - 4 (2) (2 x^{2} + 4 x + 2)}}{2 (2)} 4 y + 5 x + 5 = \pm \sqrt{25 x^{2} + 25 + 50 x - 16 x^{2} - 32 x - 16} 4 y + 5 x + 5 = \pm \sqrt{9 x^{2} + 9 - 18 x} 4 y + 5 x + 5 = \pm \sqrt{{(3 x - 3)}^{2}} 4 y + 5 x + 5 = \pm (3 x - 3) \Rightarrow 4 y + 5 x + 5 = 3 x - 3 \Rightarrow 2 x + 4 y + 8 = 0 \Rightarrow 4 y + 5 x + 5 = - (3 x - 3) \Rightarrow 8 x + 4 y + 2 = 0 (2 x + 4 y + 8) (8 x + 4 y + 2) = 0$

Equation of hyperbola having the same asmyptotes is

$(2 x + 4 y + 8) (8 x + 4 y + 2) = c$

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

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