is the image of of a reflection about the line with the equation y = x. Thus the circle in question touches the parabolae at those points where the gradient is +1.

I'll take :

Therefore the tangentpoint is

A line perpendicular to the angle bisector of the first quadrant (do say first median?) passing through T crosses this angle bisector in the center of the circle in question:

intersects y = x at

The radius is the segment