The reason it's written in terms of because the equation of a circle follows from Pythagoras' Theorem.
Do we normally avoid writing the equation of a circle as solved for to avoid the ambiguity of whether must be positive? Could we think of being negative if we describe it as a vector?
Solving directly for would force one to decide whether to put only the principal square root if they require r to be positive:
but if it can be a vector then one could put +/- correct?