Unit Circles and Parabolas

[Pre4edgc]

This may seem completely random, but my roommate and I were having an argument about whether or not there is a set of parabolas that, when paired together, a circle tangent to both of them could be a unit circle. He's positive that it's possible, but I'm not convinced, knowing that parabolas increase at different rates at different times, and a unit circle, being a constant shape and having a constant diameter, could not effectively be placed in between them for infinity.

 

Any ideas?

 

(And if you need clarifications, just ask. It seems a bit confusing, and I'm not in the state of mind right now to fix it.. Sorry! )

[ydoaPs]

y=x²+1

y=-x²-1

 

?