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, 5), radius 4

Verified step by step guidance

1. Identify the center of the circle as \((-2, 5)\) and the radius as \(4\).
2. Recall the standard form of the equation of a circle: \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
3. Substitute the center \((-2, 5)\) into the equation: \((x - (-2))^2 + (y - 5)^2 = 4^2\).
4. Simplify the equation: \((x + 2)^2 + (y - 5)^2 = 16\).
5. To graph the circle, plot the center at \((-2, 5)\) and draw a circle with radius \(4\) units 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 the center point on a coordinate plane and then using the radius to determine the circle's boundary. From the center, you can move r units in all directions (up, down, left, right) to mark points on the circle, which helps in sketching its shape accurately.

Distance Formula

The distance formula, derived from the Pythagorean theorem, calculates the distance between two points in a plane. It is given by d = √((x₂ - x₁)² + (y₂ - y₁)²). This concept is essential for understanding how far points are from the center of the circle, particularly when verifying if points lie on the circle.