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, √2), radius √2

Verified step by step guidance

1. <b>Step 1:</b> Recall 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. <b>Step 2:</b> Identify the center \((h, k)\) and the radius \(r\) from the problem. Here, the center is \((\sqrt{2}, \sqrt{2})\) and the radius is \(\sqrt{2}\).
3. <b>Step 3:</b> Substitute the values of \(h\), \(k\), and \(r\) into the standard form equation. This gives \((x - \sqrt{2})^2 + (y - \sqrt{2})^2 = (\sqrt{2})^2\).
4. <b>Step 4:</b> Simplify the equation. The right side of the equation becomes \((\sqrt{2})^2 = 2\). So, the equation of the circle is \((x - \sqrt{2})^2 + (y - \sqrt{2})^2 = 2\).
5. <b>Step 5:</b> To graph the circle, plot the center at \((\sqrt{2}, \sqrt{2})\) and draw a circle with radius \(\sqrt{2}\) 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 circular 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, which is crucial when determining if points lie on the circle.