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. insert step 1> Identify the standard form of the equation of a circle, which is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
2. insert step 2> Substitute the given center \((3, 6)\) into the equation, replacing \(h\) with 3 and \(k\) with 6.
3. insert step 3> Substitute the given radius 4 into the equation, replacing \(r\) with 4.
4. insert step 4> Square the radius to find \(r^2\), which is \(4^2 = 16\).
5. insert step 5> Write the equation of the circle using the substituted values: \((x - 3)^2 + (y - 6)^2 = 16\).

Key Concepts

Circle Equation

The standard equation of a circle in a Cartesian coordinate system is given by (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius. This formula allows us to describe all the points (x, y) that are a fixed distance (the radius) from the center point.

Center of a Circle

The center of a circle is the point from which all points on the circle are equidistant. In the equation of a circle, the center is represented by the coordinates (h, k). For the given problem, the center is (3, 6), indicating that the circle is centered at this point on the Cartesian plane.

Radius of a Circle

The radius of a circle is the distance from the center to any point on the circle. It is a crucial component in the circle's equation, as it determines the size of the circle. In this case, the radius is 4, meaning that every point on the circle is 4 units away from the center (3, 6).