The equation of a hyperbola, conjugate to the hyperbola $x^{2} + 3 x y + 2 y^{2} + 2 x + 3 y + 1 = 0$ , is

x^{2} + 3 x y + 2 y^{2} + 2 x + 3 y + 1 = 0

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x^{2} + 3 x y + 2 y^{2} + 2 x + 3 y + 2 = 0

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x^{2} + 3 x y + 2 y^{2} + 2 x + 3 y + 3 = 0

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x^{2} + 3 x y + 2 y^{2} + 2 x + 3 y + 4 = 0

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Solution

The correct option is B $x^{2} + 3 x y + 2 y^{2} + 2 x + 3 y + 2 = 0$
We know that,
Equation of hyperbola + Equation of conjugate
hyperbola = 2(Equation of asymptotes)
$∴$ Equation of conjugate hyperbola
= 2 Equation of asymptotes - Equation of hyperbola
C(x, y) = 2A(x, y) – H(x, y) …… (i)
$∵$ $H (x, y) \equiv x^{2} + 3 x y + 2 y^{2} + 2 x + 3 y = 0$
$∴$ Equation of asymptotes is
$x^{2} + 3 x y + 2 y^{2} + 2 x + 3 y + λ = 0$
$∴$ $Δ = 0, a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2} = 0, t h e n λ = 1$
$∴$ $A (x, y) \equiv x^{2} + 3 x y + 2 y^{2} + 2 x + 3 y + 1 = 0$
$∴$ $C (x, y) \equiv x^{2} + 3 x y + 2 y^{2} + 2 x + 3 y + 2 = 0$ [from equation (i)]

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