An equation can describe a curve even when one global function y(x) cannot.
On a differentiable local branch with y≠0, differentiating the constraint gives dy/dx=−x/y. The parametrization (R cosθ,R sinθ) supplies a tangent direction (−sinθ,cosθ) everywhere, including vertical tangents where the slope formula is undefined.
Implicit differentiation does not make a multivalued circle into one global function y(x).