The locus of a point whose distance from a fixed point is equal to the distance from a fixed line, which seems to be the definition of hyperbola (the fixed point is the focus and the fixed line is the directrix), so

(1) The hyperbola with the straight line x+y=2 as the directrix and the point (-1,-1) as the focus is the equation of curve C.

Then take X=0 and Y=0 respectively, and get the intersection of curve c and coordinate axis.

2) If the curve C rotates 45 clockwise around the coordinate origin, the curve C is still a hyperbola.

Only at this time, the original directrix and focus must also rotate 45 clockwise around the coordinate origin.