(1) Replace (1, 1) with y=k/x, where k= 1. The inverse analytic function is y =1/x.

(2) After rotating135, the point B' falls on the Y axis as (0, -√2), and the point A' falls on the reverse extension line of OB. The coordinate obtained by Pythagorean theorem is (-√2, -√2), so the midpoint P coordinate of A'B' is (-√2/2). Substitute into the inverse proportional function y= 1/x,

There is -√2= 1/(-√2/2)=-√2, so the equation is obviously satisfied, and point P is on this hyperbola.