In the following exercises, (a) find the center-radius form of the equation of each circle described, and (b) graph it. center (√2, √2), radius √2

Verified step by step guidance

1. Identify the given center and radius of the circle. Here, the center is \(\left(\sqrt{2}, \sqrt{2}\right)\) and the radius is \(\sqrt{2}\).
2. Recall the center-radius form of a circle's equation: \(\left(x - h\right)^2 + \left(y - k\right)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
3. Substitute the given center coordinates and radius into the formula: \(\left(x - \sqrt{2}\right)^2 + \left(y - \sqrt{2}\right)^2 = \left(\sqrt{2}\right)^2\).
4. Simplify the right side by squaring the radius: \(\left(\sqrt{2}\right)^2 = 2\), so the equation becomes \(\left(x - \sqrt{2}\right)^2 + \left(y - \sqrt{2}\right)^2 = 2\).
5. To graph the circle, plot the center at \(\left(\sqrt{2}, \sqrt{2}\right)\) on the coordinate plane, then draw a circle with radius \(\sqrt{2}\) units around this point.

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 = r², 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.

Square Roots and Simplification

Understanding how to work with square roots is essential, especially when the center coordinates and radius involve √2. Simplifying expressions and squaring these values correctly ensures accurate substitution into the circle's equation.

Graphing Circles on the Coordinate Plane

Graphing a circle requires plotting its center and using the radius to mark points in all directions. Recognizing how the radius extends from the center helps in sketching the circle accurately on the coordinate plane.