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

Verified step by step guidance

1. Recall the standard form of the equation of a circle with center \((h, k)\) and radius \(r\) is given by: \[ (x - h)^2 + (y - k)^2 = r^2 \]
2. Identify the center \((h, k)\) and radius \(r\) from the problem: here, the center is \((3, 6)\) and the radius is \(4\).
3. Substitute the values of \(h = 3\), \(k = 6\), and \(r = 4\) into the standard form equation: \[ (x - 3)^2 + (y - 6)^2 = 4^2 \]
4. Simplify the right side by squaring the radius: \[ (x - 3)^2 + (y - 6)^2 = 16 \]
5. Write the final equation of the circle as: \[ (x - 3)^2 + (y - 6)^2 = 16 \]

Key Concepts

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.

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.

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.