The equation of the hyperbola whose asymptotes are the straight lines $3 x - 4 y + 7 = 0$ and $4 x + 3 y + 1 = 0$ and
passing through the origin, is:

12 x^{2} - 7 x y - 12 y^{2} + 31 x + 17 y = 0

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12 x^{2} + 7 x y - 12 y^{2} + 31 x + 17 y = 0

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12 x^{2} - 7 x y + 12 y^{2} - 31 x + 17 y = 0

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None of these

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Solution

The correct option is A $12 x^{2} - 7 x y - 12 y^{2} + 31 x + 17 y = 0$
The combined equation of asymptotes is given by $(3 x - 4 y + 7) (4 x + 3 y + 1) = 0$
so combined equation of the hyperbola which differ by a constant is given by
$(3 x - 4 y + 7) (4 x + 3 y + 1) + λ = 0$ it passes through origin
$∴ 7 + λ = 0 \Rightarrow λ = - 7$
$∴$ equation of hyperbola is
$(3 x - 4 y + 7) (4 x + 3 y + 1) - 7 = 0 i . e . 12 x^{2} - 7 x y - 12 y^{2} + 31 x + 17 y = 0$
Hence, option 'A' is correct.

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