Assuming the focii lie on $x$-axis and the transverse axis is $y=0$ (i.e. $x-axis$).
So, distance between the two focii of the hyperbola is given by : $2c=16$ $\Rightarrow$ $c=8$
Also, eccentricity of hyperbola is $e=\frac{c}{a}$ $\Rightarrow$ $a=\frac{8}{\sqrt{2}}$ $\Rightarrow$ $a^{2}=32$
Also, for hyperbola $c^{2}=a^{2}+b^{2}$ $\Rightarrow$ $b^{2}=64-\frac{64}{2}=32$
So, the equation of hypebola would be : $\frac{x^{2}}{32}-\frac{y^{2}}{32}=1$ $\Rightarrow$ $x^{2}-y^{2}=32$
Option D is correct.
