Let the maximum radius be r, and the coordinates of the tangent point between the ball and the right side of the parabola be (a, b).

As shown in the figure, b=4-r can be obtained.

According to Pythagorean theorem, a? =r？ -2? =r？ -4

Because b=a? /4,

So 4-r=a? /4=r？ /4- 1

The solution is r = 2 √ 6-2.

Discard negative roots, with a maximum radius of 2√6-2.