Find the equation to the hyperbola,conjugate to the hyperbola $2 x^{2} + 3 x y - 2 y^{2} - 5 x + 5 y + 2 = 0$ .

2 x^{2} + 3 x y - 2 y^{2} - 5 x + 5 y - 8 = 0

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x^{2} + 3 x y - y^{2} - 5 x + 5 y - 8 = 0

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x^{2} + 3 x y - y^{2} - 5 x + 5 y + 8 = 0

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Solution

The correct option is A $2 x^{2} + 3 x y - 2 y^{2} - 5 x + 5 y - 8 = 0$
Let Asymptotes :
$2 x^{2} + 3 x y - 2 y^{2} - 5 x + 5 y + λ = 0$ .
$∴ a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2} = 0$
$λ = - 5$
Equation Hyperbola + Conjugate Hyperbola $=$ 2 (Asympototes)
$∴$ Conjugate Hyperbola $=$ 2 (Asymptotes) $-$ Hyperbola
Therefore, equation of conjugate hyperbola is $2 x^{2} + 3 x y - 2 y^{2} - 5 x + 5 y - 8 = 0$

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