find radius of maximum area that lies between the two circles and commom tangent AB!?

The center of the new circle must be equidistant from both circles and the common tangent.
x²+(y-3)²=(3+r)² and (x-2√3)²+(y-1)^2=(1+r)² and y=r

which you can resolve to: x = 3√3-3, y = (6-3√3)/2 = radius ← ANSWER

A radius is a line and does not have area, so Euclid
states.