Derive the equation of a circle from center/radius and complete the square to identify center/radius.
Complete a circle equation and classify a point consistently
Problem
For \(x^{2}~+~y^{2}~-~6x~+~4y~-~3~=~0\) and \(P~=~(7,~-2)\), complete \(\text{both}~\text{squares}\) to find center-radius form. Report the center and radius, compute \(P\)'s squared center distance, compare it with \(r^{2}\), and classify \(P\).
Big Picture
What this problem is really about
One consistent circle record should drive every part of the classification. We’ll complete and balance both variable squares, factor to reveal the center and radius squared, compute the point’s squared center distance, and compare squared quantities so the equation, geometry, and point position cannot contradict one another.
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