The equation of a hyperbola, conjugate to the hyperbola $x^{2} + 3 x y + 2 y^{2} + 2 x + 3 y = 0$ 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 $x^{2} + 3 x y + 2 y^{2} + 2 x + 3 y + 2 = 0$
$H : x^{2} + 3 x y + 2 y^{2} + 2 x + 3 y = 0$
Let the pair of asympt at is be
$A : x^{2} + 3 x y + 2 y^{2} + 2 x + 3 y + k = 0$
$∴$ It represent a apir of straight lines, satisfying the condition :
$a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2} = 0$
$\Rightarrow 1 \times 2 \times k + 2 (\frac{3}{2}) \times 1 \times (\frac{3}{2}) - 1 {(\frac{3}{2})}^{2} - 2 (1)^{2} - k {(\frac{3}{2})}^{2} = 0$
$\Rightarrow 2 k + \frac{9}{4} - 2 - \frac{9}{4} k = 0 \Rightarrow k = 1$
$\Rightarrow A^{2} . x^{2} + 3 x y + 2 y^{2} + 2 x + 3 y + 1 = 0$
As $H + C = 2 A \Rightarrow C$ (conjugate $= 2 A - H$ hypergate)
$\Rightarrow C : x^{2} + 3 x y + 2 y^{2} + 2 x + 3 y + 2 = 0$

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