If it's tangent to all four of the circles, then the derivatives will be equal at the points where they touch.
should be the required circle.
this is because the radius of each circle is 5 units and their centres are at (-5,0), (5,0), (0,5) and (0,-5).
if you draw a rough figure its easy to see that a symmetric flower shaped digram appears. and then its easy to find the required circle.
In general a circle tangent to four given circles will not exist. In this special symmetric situation it exists.