Derive the equation of a circle from center/radius and complete the square to identify center/radius.
Derive a circle equation from the distance formula
Problem
Let \((x,~y)\) be a point on the circle with center \((0,~0)\) and radius \(5\). Write its point-to-center distance equation, square \(\text{both}~\text{sides}\), and simplify to the circle equation.
A circle equation grows directly from its fixed-distance definition. We’ll express the distance from a general point to the center, set that distance equal to the radius, square both nonnegative sides, simplify the center shifts, and verify the resulting locus with a known point on the boundary.
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