Consider two parabolas
Find the equation of circle of minimum radius touching both the parabolas.
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
2. If a circle and a parabola are tangent to each other then they have a common tangent. This tangent must be perpendicular to the radius of the circle. The radius is perpendicular to the angle bisector and thus the tangent must be parallel to the angle bisector. That means the slope of the tangent is equal to the slope of the parabola (in the tangent point) and it is equal to the slope of th angle bisector.
3. Since I knew the slope of the angle bisector I used this fact to calculate the coordinates of the tangent point at the parabola.