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

Name: Trigonometry
Author: Lial
ISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891

검증된 단계별 안내

Identify the center and radius of the circle. Here, the center is given as \(\left(\sqrt{2}, \sqrt{2}\right)\) and the radius is \(\sqrt{2}\).

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.

Substitute the given center coordinates and radius into the formula: \(\left(x - \sqrt{2}\right)^2 + \left(y - \sqrt{2}\right)^2 = \left(\sqrt{2}\right)^2\).

Simplify the right side of the equation by squaring the radius: \(\left(\sqrt{2}\right)^2 = 2\), so the equation becomes \(\left(x - \sqrt{2}\right)^2 + \left(y - \sqrt{2}\right)^2 = 2\).

For graphing, 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.

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 relates the coordinates of any point on the circle to its center and radius, making it easy to write and analyze the circle's equation.

Understanding Coordinates and Radicals

Coordinates like (√2, √2) involve square roots, which are irrational numbers. Recognizing how to handle these values in algebraic expressions and graphing is essential, as they affect the exact position of the circle's center on the Cartesian plane.

Graphing Circles on the Cartesian Plane

Graphing a circle requires plotting its center and using the radius to mark points at a fixed distance in all directions. Understanding how to translate the center-radius form into a visual representation helps in accurately sketching the circle.