Derive the equation of a circle from center/radius and complete the square to identify center/radius.
Interpret a contextual circle boundary and covered region
Problem
A coverage map uses coordinates in miles and has boundary equation \((x~-~2)^{2}~+~(y~-~3)^{2}~=~25\). Identify the coverage center and reach, explain what equality represents, and write the inequality for every covered point, including the inside, boundary, and outside tests.
Big Picture
What this problem is really about
The equation with equality describes only the coverage boundary, while the context asks about the whole covered region. We’ll read the source location and linear reach, distinguish radius from radius squared, replace equality with the inclusive distance condition, and classify interior, boundary, and exterior points by squared distance.
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