This problem mainly examines the synthesis of circles. The key is to find out the relationship between line segments by combining the knowledge of circle with that of congruent triangles and similar triangles. This is more difficult. You have to think about it and think about the answer. /Exercise/Math /799705 Ask me if you don't understand. I hope I can help you. I wish you progress in your study and will definitely adopt it. Thank you. Let's go

In the plane rectangular coordinate system xOy, O is the coordinate origin, the circle P centered on P( 1, 1) is tangent to point M and point N respectively, and point F starts from point M and moves at the speed of 1 unit length per second along the positive direction of X axis, connecting PF, and the intersection point PE is perpendicular to the intersection point of PF and Y axis at point E, and a point is set.

(2) During the movement of point F, let OE=a and OF=b, and try to represent b with an algebraic expression containing a;

(3) Let point F be the symmetrical point {F}' about point M, and connect QE by intersecting the parabola symmetry axes of three points M, E and {F}' with point Q.. Is there a moment in the movement of point F, which makes the triangle with points Q, O and E similar to the triangle with points P, M, F, M and F? If it exists, write the value of t;