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