A line segment through the center of each circle intersects the circle at the points shown. a. Find the coordinates of the circle's center. b. Find the radius of the circle. c. Use your answers from parts (a) and (b) to write the standard form of the circle's equation.

검증된 단계별 안내

Identify the two points on the circle that the line segment passes through. For the first circle, these points are (-5, 9) and (-3, 5). For the second circle, the points are (3, 6) and (5, 4).

Find the center of the circle by calculating the midpoint of the line segment connecting the two points. Use the midpoint formula: \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\).

Calculate the radius of the circle by finding the distance from the center to either of the two points on the circle. Use the distance formula: \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\), then divide by 2 since the segment is a diameter.

Write the standard form of the circle's equation using the center \((h, k)\) and radius \(r\): \[(x - h)^2 + (y - k)^2 = r^2\].

Substitute the values of the center coordinates and the radius squared into the standard form equation to express the equation of the circle.

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주요 개념

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

Midpoint Formula

The midpoint formula finds the center point between two given points by averaging their x-coordinates and y-coordinates. It is essential here to find the circle's center since the line segment passes through the center and connects two points on the circle.

Distance Formula

The distance formula calculates the length between two points in the coordinate plane. It helps determine the radius of the circle by finding the distance from the center to one of the points on the circle.

Standard Form of a Circle's Equation

The standard 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. Using the center and radius found, this formula expresses the circle algebraically.