In the following exercises, (a) find the center-radius form of the equation of each circle described, and (b) graph it. center (5, -4), radius 7

Verified step by step guidance

1. Identify the given information: the center of the circle is at the point \( (5, -4) \) and the radius is \( 7 \).
2. Recall the center-radius form of a circle's equation: \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center and \( r \) is the radius.
3. Substitute the center coordinates \( h = 5 \) and \( k = -4 \) into the formula: \( (x - 5)^2 + (y - (-4))^2 = r^2 \).
4. Simplify the expression inside the parentheses: \( (x - 5)^2 + (y + 4)^2 = r^2 \).
5. Substitute the radius \( r = 7 \) and square it to get \( r^2 = 49 \), so the equation becomes \( (x - 5)^2 + (y + 4)^2 = 49 \). This is the center-radius form of the circle.

Key Concepts

Equation of a Circle in Center-Radius Form

The center-radius form of a circle's equation is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius. This form directly shows the circle's location and size, making it easier to graph and analyze.

Identifying the Center and Radius

Given the center coordinates (h, k) and radius r, you substitute these values into the center-radius formula. Understanding how to correctly place the center values with opposite signs inside the parentheses is essential for writing the equation accurately.

Graphing a Circle

To graph a circle, plot the center point first, then use the radius to mark points in all directions (up, down, left, right) from the center. Connecting these points smoothly forms the circle, helping visualize its size and position on the coordinate plane.