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 (5, -4), radius 7

Verified step by step guidance

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. Substitute the given center \((5, -4)\) into the equation, replacing \(h\) with 5 and \(k\) with -4.
3. Substitute the given radius 7 into the equation, replacing \(r\) with 7.
4. Write the equation in the center-radius form: \((x - 5)^2 + (y + 4)^2 = 49\).
5. To graph the circle, plot the center at \((5, -4)\) on the coordinate plane, and draw a circle with radius 7 units around this 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 its center on a coordinate plane and using the radius to determine the circle's boundary. From the center, you can measure the radius in all directions to mark points on the circle, which helps visualize its shape and size.

Coordinate Geometry

Coordinate geometry, or analytic geometry, involves using a coordinate system to represent geometric shapes and their properties. Understanding how to manipulate and interpret coordinates is essential for solving problems related to circles, including finding their equations and graphing them accurately.