Derive the equation of a circle from center/radius and complete the square to identify center/radius.
Classify a circle equation by the sign of radius squared
Problem
Classify the real locus of \(x^{2}+y^{2}-6x+4y-3=0\).
Big Picture
What this problem is really about
Not every completed-square equation describes a positive-radius real circle, so the right side needs a sign check. We’ll convert to center-radius form, identify the center and radius-squared value, classify the locus as a circle, single point, or empty set from that sign, and only then recover a radius when appropriate.
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