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 (5, -4), radius 7

Verified step by step guidance

1. Identify the center of the circle as \((h, k) = (5, -4)\) and the radius \(r = 7\).
2. Recall the center-radius form of a circle's equation: \((x - h)^2 + (y - k)^2 = r^2\).
3. Substitute the center \((h, k) = (5, -4)\) and radius \(r = 7\) into the equation.
4. Write the equation as \((x - 5)^2 + (y + 4)^2 = 49\).
5. To graph the circle, plot the center at \((5, -4)\) and draw a circle with radius 7 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, interpret equations, and visualize shapes in this system is essential for solving problems related to circles and other geometric figures.