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(B) The distance between the foci of a hyperbola is given by $2ae = 16$.Given eccentricity $e = \sqrt{2}$,we have $2a(\sqrt{2}) = 16$,which simplifies

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$x^2 - y^2 = 32$

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(B) The distance between the foci of a hyperbola is given by $2ae = 16$.Given eccentricity $e = \sqrt{2}$,we have $2a(\sqrt{2}) = 16$,which simplifies to $a = \frac{8}{\sqrt{2}} = 4\sqrt{2}$.For a hyperbola,$b^2 = a^2(e^2 - 1)$.Substituting the values,$b^2 = (4\sqrt{2})^2 ((\sqrt{2})^2 - 1) = 32(2 - 1) = 32$.Since $a^2 = 32$ and $b^2 = 32$,the hyperbola is a rectangular hyperbola with the equation $x^2 - y^2 = a^2$,which is $x^2 - y^2 = 32$.

The equation of the hyperbola referred to the coordinate axes as axes of symmetry,whose distance between the foci is $16$ and eccentricity is $\sqrt{2}$,is

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