Equation of the rectangular hyperbola whose focus is $(1, 1)$ and the corresponding directrix $x - y + 1 = 0$ is

x^{2} - y^{2} = 1

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x y = 1

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2 x y - 4 x + 4 y + 1 = 0

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2 x y + 4 x - 4 y - 1 = 0

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Solution

The correct option is C $2 x y - 4 x + 4 y + 1 = 0$
Required hyperbola is given by,
$(x - 1)^{2} + (y + 1)^{2} = (x - y + 1)^{2}$
$\Rightarrow x^{2} - 2 x + 1 + y^{2} + 2 y + 1 = x^{2} + y^{2} + 1 - 2 x y - 2 y + 2 x$
$\Rightarrow 2 x y - 4 x + 4 y + 1 = 0$

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