The correct options are
A Eccentricity e=√2
B Coordinates of foci are (±2√2,±2√2) and equation of the directrix is given by x+y±2√2=0
D Length of latus rectum =4√2
For a rectangular hyperbola of the form xy=c2 , the equation of the traverse axis is y=x.
Hence, option A is incorrect.
The eccentricity is √2. Hence, option B is correct.
The coordinates of foci are (c√2,c√2) . For this hyperbola, c=2. Hence, the coordinates of foci are (2√2,2√2). Hence, option C is correct.
The length of the latus rectum is given by 2c√2. Hence, the length of the latus rectum 4√2. Hence, option D is also correct.