Question

A circle centered at P with radius 4 cm and another centered at Q with radius 16 cm touch each other externally. A third circle with centre R is drawn to touch the first two circles. they all have a common tangent. Then the radius of the circle with centre R is

Answer 1

There are infinite circles with center R and with any radius.
See diagram.

Given two circles are 5 and 6.

The blue circles  numbered 1 to 4 are all possible solutions.  These are infinite in number.  They are within circle 5 or circle 6.  Also they are outside circle 5 and outside circle 6.

The centers of all circles  ie., P, Q and R are collinear.

obviously the common tangent is perpendicular to the line joining all centers of all circles.
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Also you find two more circles  numbered 7 and 8.,  These are tangentially touching circles 5 and 6, as well as the common tangent of circles 5 and 6.