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

Axis of parabola is y = x
Since vertex is at a  distance of $\sqrt{2}$ from (0, 0), vertex is
A(1, 1), also, focus is at a  distance of  $2\sqrt{2}$ from (0, 0),focus is S(2, 2). Distance SA =$\sqrt{2}$ .
So, directrix is at  a distance $\sqrt{2}$ from origin, which is x + y = 0. Hence equation of the parabola is $\sqrt{(\mathrm{x}-2{)}^2+(\mathrm{y}-2{)}^2}=\frac{|\mathrm{x}+\mathrm{y}|}{\sqrt{2}}(\because \mathrm{SP}=\mathrm{PM})$
 $\Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{xy}=8(\mathrm{x}+\mathrm{y}-2)\text{ ⇒ }(\mathrm{x}-\mathrm{y}{)}^2=8(\mathrm{x}+\mathrm{y}-2)$