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Question

If the equation to the hyperbola is 3x2−5xy−2y2+5x+11y−8=0 then the equation of conjugate hyperbola and pair of asymptotes is

A
Conjugate Hyperbola: 3x25xy2y2+5x+11y16=0
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B
Asymptotes: 3x25xy2y2+5x+11y12=0
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C
Asymptotes: 3x25xy2y2+5x+11y+12=0
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D
Conjugate Hyperbola: 3x25xy2y2+5x+11y24=0
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Solution

The correct option is B Asymptotes: 3x25xy2y2+5x+11y12=0
Given: Hyperbola 3x25xy2y2+5x+11y8=0

To Find: Equation of asymptotes and conjugate hyperbola

3x25xy2y2+5x+11y8=0

Equation of asymptotes is 3x25xy2y2+5x+11y+c=0

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

=abc+2fghaf2bg2ch2=0

a=3,h=52,b=2,g=52,f=112

=(3)(52)c+2(112)(52)(52)3(112)2(2)(52)2c(52)2=0

=6c27543634+50425c4=0

c=58849=12

Asymptote equation is 3x25xy2y2+5x+11y12=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.

3x25xy2y2+5x+11y8+conjugate hyperbola=2(3x25xy2y2+5x+11y12)

conjugate hyperbola=3x25xy2y2+5x+11y16=0

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