The equation of a circle can be derived using the Pythagorean theorem. The standard form of the equation of a circle with its center at the point (h, k) and radius r is:

\[ (x – h)^2 + (y – k)^2 = r^2 \]

Derivation of the Formula

Distance Formula: The distance between any point (x, y) on the circle and the center (h, k) is the radius r. The distance formula between two points (x1, y1) and (x2, y2) is:

\[√((x2 – x1)^2 + (y2 – y1)^2)\]

1. 1. Using the Distance Formula: Substituting the points (x, y) and (h, k) into the distance formula gives us:

\[√((x – h)^2 + (y – k)^2) = r\]

1. 1. Squaring Both Sides: To eliminate the square root, we square both sides of the equation:

\[(x – h)^2 + (y – k)^2 = r^2\]

This equation represents all the points (x, y) that are r units away from the center (h, k). It defines the boundary of the circle.

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