The axis of a parabola is along the line $y=x$and the distance of its vertex and focus from the origin are $\sqrt{2}$and $2\sqrt{2}$, respectively. If vertex and focus both lie in the first quadrant, then the equation of the parabola is ${\left(x+y\right)}^2=\left(x-y-2\right)$ ${\left(x-y\right)}^2=\left(x+y-2\right)$ ${\left(x-y\right)}^2=4\left(x+y-2\right)$ ${\left(x-y\right)}^2=8\left(x+y-2\right)$

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