First we find the points of intersection of the circle and
parabola by solving the system:

 

We get this solution:





Since the circle touches the parabola in 2 places, like this:

 

the value of c must be greater than it is in the case 
where all four solutions coincide and are equal at the origin (0,0). 
That is the case when y=0, and r=c

 



To find c in that case, we setting y=0 and r=c



















That is the case when c=r=1/2, therefore we must have c>1/2.

Edwin