Do we normally avoid writing the equation of a circle as solved for $\displaystyle r$ to avoid the ambiguity of whether $\displaystyle r$ must be positive? Could we think of $\displaystyle r$ being negative if we describe it as a vector?

Starting with:

$\displaystyle (x-h)^2+(y-k)^2=r^2$

Solving directly for $\displaystyle r$ would force one to decide whether to put only the principal square root if they require r to be positive:

$\displaystyle r=\sqrt{(x-h)^2+(y-k)^2}$

but if it can be a vector then one could put +/- correct?

$\displaystyle r=\pm\sqrt{(x-h)^2+(y-k)^2}$