CONCEPT PREVIEW Fill in the blank(s) to correctly complete each sentence. The circle with center (3, 6) and radius 4 has equation _________.

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Recall the standard form of the equation of a circle with center \((h, k)\) and radius \(r\) is given by: \[ (x - h)^2 + (y - k)^2 = r^2 \]

Identify the center \((h, k)\) and radius \(r\) from the problem: here, the center is \((3, 6)\) and the radius is \(4\).

Substitute the values of \(h = 3\), \(k = 6\), and \(r = 4\) into the standard form equation: \[ (x - 3)^2 + (y - 6)^2 = 4^2 \]

Simplify the right side by squaring the radius: \[ (x - 3)^2 + (y - 6)^2 = 16 \]

Write the final equation of the circle as: \[ (x - 3)^2 + (y - 6)^2 = 16 \]

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Equation of a Circle

The equation of a circle with center (h, k) and radius r is given by (x - h)² + (y - k)² = r². This formula represents all points (x, y) that are exactly r units away from the center.

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Equations of Circles & Ellipses

Coordinates of the Center

The center of the circle is a fixed point (h, k) from which every point on the circle is equidistant. Identifying the center coordinates is essential to correctly substitute values into the circle's equation.

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Intro to Polar Coordinates

Radius of the Circle

The radius is the distance from the center to any point on the circle. Squaring the radius (r²) is necessary in the equation to express the set of points forming the circle.

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