If the equation to the hyperbola is $3 x^{2} - 5 x y - 2 y^{2} + 5 x + 11 y - 8 = 0$ then the equation of conjugate hyperbola and pair of asymptotes is

Conjugate Hyperbola:

3 x^{2} - 5 x y - 2 y^{2} + 5 x + 11 y - 16 = 0

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Asymptotes:

3 x^{2} - 5 x y - 2 y^{2} + 5 x + 11 y - 12 = 0

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Asymptotes:

3 x^{2} - 5 x y - 2 y^{2} + 5 x + 11 y + 12 = 0

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Conjugate Hyperbola:

3 x^{2} - 5 x y - 2 y^{2} + 5 x + 11 y - 24 = 0

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Solution

The correct option is B Asymptotes: $3 x^{2} - 5 x y - 2 y^{2} + 5 x + 11 y - 12 = 0$
Given: Hyperbola $3 x^{2} - 5 x y - 2 y^{2} + 5 x + 11 y - 8 = 0$

To Find: Equation of asymptotes and conjugate hyperbola

$3 x^{2} - 5 x y - 2 y^{2} + 5 x + 11 y - 8 = 0$

Equation of asymptotes is $3 x^{2} - 5 x y - 2 y^{2} + 5 x + 11 y + c = 0$

Since, it's a pair of straight lines, the determinant value will be zero.

$△ = a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2} = 0$

$a = 3, h = - \frac{5}{2}, b = - 2, g = \frac{5}{2}, f = \frac{11}{2}$

$\Rightarrow △ = (3) (- \frac{5}{2}) c + 2 (\frac{11}{2}) (\frac{5}{2}) (- \frac{5}{2}) - 3 (\frac{11}{2})^{2} - (- 2) (\frac{5}{2})^{2} - c (- \frac{5}{2})^{2} = 0$

$\Rightarrow △ = - 6 c - \frac{275}{4} - \frac{363}{4} + \frac{50}{4} - \frac{25 c}{4} = 0$

$\Rightarrow c = - \frac{588}{49} = - 12$

Asymptote equation is $3 x^{2} - 5 x y - 2 y^{2} + 5 x + 11 y - 12 = 0$

We know that, the equation of asymptote differs by the equation of hyperbola by some constant. It will differ with the conjugate hyperbola equation by same constant value.

$3 x^{2} - 5 x y - 2 y^{2} + 5 x + 11 y - 8 + conjugate hyperbola = 2 (3 x^{2} - 5 x y - 2 y^{2} + 5 x + 11 y - 12)$

$conjugate hyperbola = 3 x^{2} - 5 x y - 2 y^{2} + 5 x + 11 y - 16 = 0$

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