1. Introduction and Key Concept: Symmetry and Smallest Circle

The problem asks for the radius of the smallest circle that touches two parabolas: $P_1: y=x^2+2$ and $P_2: x=y^2+2$.

• Symmetry is Key: Observe the equations. If we swap $x$ and $y$ in $P_1$'s equation, we get $P_2$'s equation. This means the two parabolas are reflections of each other across the line $y=x$.
• Smallest Circle Condition: For the smallest circle touching two curves that are symmetric with respect to a line, the center of the circle must lie on that line of symmetry. In this case, the center of