In the following exercises, (a) find the center-radius form of the equation of each circle described, and (b) graph it. See Examples 1 and 2. center (2, 0), radius 6

Verified step by step guidance

1. Identify the center and radius of the circle from the problem statement. The center is given as (2, 0) and the radius is 6.
2. Recall the standard form of the equation of a circle with center (h, k) and radius r: \((x - h)^2 + (y - k)^2 = r^2\).
3. Substitute the center coordinates (h, k) = (2, 0) and the radius r = 6 into the standard form equation.
4. Simplify the equation by squaring the radius and writing out the complete equation.
5. To graph the circle, plot the center at (2, 0) on a coordinate plane, and use the radius of 6 units to draw a circle around the center, ensuring all points on the circle are 6 units away from the center.

Key Concepts

Center-Radius Form of a Circle

The center-radius form of a circle's equation is expressed as (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius. This format allows for easy identification of the circle's center and radius, facilitating both graphing and analysis.

Graphing a Circle

Graphing a circle involves plotting the center point on a coordinate plane and then using the radius to determine the circle's boundary. From the center, you can measure the radius in all directions to mark points that define the circle, which is then drawn as a smooth curve connecting these points.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is the study of geometric figures using a coordinate system. It combines algebra and geometry, allowing for the representation of shapes like circles, lines, and polygons in a two-dimensional space, which is essential for solving problems involving their equations and graphs.