Find the asymptotes of the hyperbola $2 x^{2} - 3 x y - 2 y^{2} + 3 x - y + 8 = 0$ . Also find the equation to the conjugate hyperbola & the equation of the principal axes of the curve.

x - 2 y + 1 = 0; 2 x + y + 1 = 0; 2 x^{2} - 3 x y - 2 y^{2} + 3 x - y - 6 = 0; 3 x y + 2 = 0; x - 3 y = 0

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x + 2 y - 1 = 0; 2 x + y + 1 = 0; 2 x^{2} - 3 x y - 2 y^{2} + 3 x - y + 6 = 0; 3 x y + 2 = 0; x + 3 y = 0

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x - 2 y + 1 = 0; 2 x + y + 1 = 0; 2 x^{2} - 3 x y - 2 y^{2} + 3 x - y - 6 = 0; 3 x y + 2 = 0; x + 3 y = 0

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x - 2 y + 1 = 0; 2 x - y + 1 = 0; 2 x^{2} - 3 x y - 2 y^{2} + 3 x - y + 6 = 0; 3 x y - 2 = 0; x - 3 y = 0

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Solution

The correct option is C $x - 2 y + 1 = 0; 2 x + y + 1 = 0; 2 x^{2} - 3 x y - 2 y^{2} + 3 x - y - 6 = 0; 3 x y + 2 = 0; x + 3 y = 0$
Let equation of asymptotes are
$2 x^{2} - 3 x y - 2 y^{2} + 3 x - y + 8 + λ = 0$
As it represents two straight lines
$∴ - 4 (8 + λ) + \frac{9}{4} - \frac{1}{2} + \frac{9}{2} - (8 + λ) \frac{9}{4} = 0$
$\Rightarrow λ = - 7$
So asymptotes are $2 x^{2} - 3 x y - 2 y^{2} + 3 x - y + 1 = 0$
$\Rightarrow$ 2y - x - 1 = 0 & 2x + y + 1 = 0
and the equation of conjugate hyperbola will be
$2 x^{2} - 3 x y - 2 y^{2} + 3 x - y + 8 - 14 = 0$ .

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A

20 A

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