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, √2), radius √2

Verified step by step guidance

1. Identify the center of the circle as \((\sqrt{2}, \sqrt{2})\) and the radius as \(\sqrt{2}\).
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 \((h, k) = (\sqrt{2}, \sqrt{2})\) and the radius \(r = \sqrt{2}\) into the equation.
4. The equation becomes \((x - \sqrt{2})^2 + (y - \sqrt{2})^2 = (\sqrt{2})^2\).
5. Simplify the equation to \((x - \sqrt{2})^2 + (y - \sqrt{2})^2 = 2\).

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 of the circle's properties.

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 can then be connected to form the circular shape.

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, which helps in verifying if points lie on the circle.