If the shortest distance of a given point to a given circle is 4 cm and the longest distance is 9 cm, then the radius of the circle is

Lets assume the distance b/w point and centre of circle as "d" . let radius of circle be R .

There are two cases , one case is in which point is inside the circle and the other one is point lies totally outside circle :

Case I : Point lying outside the circle 
  

The shortest distance is PB = PO - BO = d - r = 4
The longest distance is PA = PO + OA = d + r = 9
Solving this we get r = 2.5 .

Case II : Point lying inside the circle 

The shortest distance is PB = OB - OP = r - d = 4
The longest distance is PA = PO + OA = r + d  = 9
Solving this we get r = 6.5

Therefore the radius of circle can be either 2.5 cm or 6.5 cm