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, 5), radius 4

Verified step by step guidance

1. Step 1: Recall 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 of the circle and \(r\) is the radius.
2. Step 2: Substitute the given center \((-2, 5)\) into the equation. This means \(h = -2\) and \(k = 5\).
3. Step 3: Substitute the given radius \(4\) into the equation. This means \(r = 4\).
4. Step 4: Write the equation of the circle using the values from Steps 2 and 3: \((x + 2)^2 + (y - 5)^2 = 16\).
5. Step 5: To graph the circle, plot the center at \((-2, 5)\) and draw a circle with radius \(4\) around this point.

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 its center on a coordinate plane and using the radius to determine the points that lie on the circle. From the center, you can move r units in all directions (up, down, left, right) to find key points, which helps in sketching the circle accurately.

Coordinate Geometry

Coordinate geometry is the study of geometric figures using a coordinate system, typically the Cartesian plane. Understanding how to plot points, lines, and curves in this system is essential for visualizing and solving problems related to circles and other shapes.