In the following exercises, (a) find the center-radius form of the equation of each circle described, and (b) graph it. See Examples 5 and 6.
center (2, 0), radius 6

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Recall that the center-radius form of a circle's equation is given by \[(x - h)^2 + (y - k)^2 = r^2\] where \[(h, k)\] is the center and \[r\] is the radius.

Identify the center coordinates and radius from the problem: the center is \[(2, 0)\] and the radius is \[6\].

Substitute the center coordinates into the formula: replace \[h\] with \[2\] and \[k\] with \[0\], so the equation becomes \[(x - 2)^2 + (y - 0)^2 = r^2\].

Substitute the radius value into the equation: replace \[r\] with \[6\], so the equation becomes \[(x - 2)^2 + y^2 = 6^2\].

Simplify the right side by squaring the radius: the equation is \[(x - 2)^2 + y^2 = 36\]. This is the center-radius form of the circle's equation.

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The center-radius form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. This form directly shows the circle's location and size, making it easy to write the equation when the center and radius are known.

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The radius is the fixed distance from the center to any point on the circle. Knowing how to interpret and use this distance helps in both writing the equation and graphing the circle accurately, ensuring all points satisfy the distance condition.

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