Derive the equation of a circle from center/radius and complete the square to identify center/radius.
Write a circle equation from its center and a point on the circle
Problem
A circle has center \((2,~1)\) and passes through \((5,~5)\). Calculate the squared radius from the coordinate differences, then write the center-radius equation.
The given point is not another center; it supplies the center-to-boundary displacement. We’ll place the known center into standard form, subtract coordinates to get the horizontal and vertical changes, use their squared sum directly as radius squared, and substitute the point back to verify the circle.
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