The distance of vertex from the directrix is $\sqrt 2$
Further, the slope of the axis is $-1$
The focus is $(h,k)$
Then $$\frac{h-3}{\frac{-1}{\sqrt 2}}=\sqrt 2$$ $$h=2$$
And $$\frac{k-2}{\frac{1}{\sqrt 2}}=\sqrt 2$$ $$k=3$$
The slope of latus rectum is 1
$$y-3=x-2$$$$x-y=-1$$
The answer given is $x-y=3$
What’s going wrong?
$h$ and $k$ are not computed correctly. They should be,
$$h= 3+ \frac1{\sqrt2}\sqrt2=4,\>\>\>\>k=2-\frac1{\sqrt2}\sqrt2=1$$
which yields $x-y=3$.