The equation of the hyperbola whose asymptotes are $2 x - y = 3$ and $3 x + y = 7$ and passing though the point $(1, 1)$ is

6 x^{2} - x y - y^{2} - 20 x + 4 y + 12 = 0

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6 x^{2} - x y - y^{2} - 23 x + 4 y + 15 = 0

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6 x^{2} - x y - y^{2} + 20 x - 4 y - 20 = 0

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6 x^{2} + x y + y^{2} - 23 x - 4 y + 19 = 0

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Solution

The correct option is B $6 x^{2} - x y - y^{2} - 23 x + 4 y + 15 = 0$
The combined equation of the asymptotes is $(2 x - y - 3) (3 x + y - 7) = 0$
$\Rightarrow 6 x^{2} - x y - y^{2} - 23 x + 4 y + 21 = 0$

Let the equation of the hyperbola be $6 x^{2} - x y - y^{2} - 23 x + 4 y + λ = 0,$

where $λ$ is a constant such that it represents hyperbola which passes through $(1, 1)$ ,
so $λ = 15.$

Hence, the equation of the hyperbola becomes $6 x^{2} - x y - y^{2} - 23 x + 4 y + 15 = 0.$

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2 x - y = 3

3 x + y = 7

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3 x + y = 7

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The equation of the hyperbola whose asymptotes are the lines 3x – 4y + 7 = 0 and 4x + 3y + 1 = 0 and which passes through origin is

x = 3

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