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 (0, 0), radius 6

검증된 단계별 안내

Identify the general form of the equation of a circle with center at \((h, k)\) and radius \(r\), which is given by: \[ (x - h)^2 + (y - k)^2 = r^2 \]

Since the center is \((0, 0)\), substitute \(h = 0\) and \(k = 0\) into the formula, simplifying it to: \[ x^2 + y^2 = r^2 \]

Given the radius \(r = 6\), substitute this value into the equation: \[ x^2 + y^2 = 6^2 \]

Simplify the right side of the equation to express the center-radius form: \[ x^2 + y^2 = 36 \]

For graphing, plot the center at the origin \((0, 0)\) and draw a circle with radius 6 units extending equally in all directions from the center.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

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 = r^2, 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.

Identifying the Center and Radius

To write the equation of a circle, you must know its center coordinates (h, k) and radius r. For a circle centered at (0, 0) with radius 6, the equation simplifies to x^2 + y^2 = 36, since h and k are zero and r^2 = 36.

Graphing a Circle

Graphing a circle involves plotting its center point and drawing all points at a distance equal to the radius from the center. For a circle centered at the origin with radius 6, plot (0,0) and mark points 6 units away in all directions to sketch the circle.