A circle in the xy-plane has its center on the line y=1. If the point (2,-3) lies on the circle and the radius is 4 , which of the following could be the center of the circle? Choose 1 answer: (A) (2,1) (B) (2,-3) (c) (4,1) (D) (-4,1)

Question

A circle in the  xy-plane has its center on the line  y=1. If the point  (2,-3) lies on the circle and the radius is 4 , which of the following could be the center of the circle? Choose 1 answer: (A)  (2,1) (B)  (2,-3) (c)  (4,1) (D)  (-4,1)

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Problem and Given Information: Understand the problem and the given information. We are given that the center of the circle lies on the line y=1, which means that the y-coordinate of the center is 1. We also know that the circle passes through the point (2,-3) and has a radius of 4 units.Distance Formula for x-coordinate: Use the distance formula to find the possible x-coordinates of the center. The distance formula is d = √((x2 - x1)² + (y2 - y1)²), where d is the distance between two points (x1, y1) and (x2, y2). Here, we know the distance (radius) is 4, one point is (2,-3), and the other point (the center) has a y-coordinate of 1. We need to find the x-coordinate of the center.Solving for x-coordinate: Plug in the known values into the distance formula and solve for the x-coordinate. Let the x-coordinate of the center be x. Then we have: 4 = √((x - 2)² + (1 - (-3))²) 16 = (x - 2)² + (1 + 3)² 16 = (x - 2)² + 16 0 = (x - 2)² x - 2 = 0 x = 2Checking Possible Centers: Check the possible centers against the given options. We have found that the x-coordinate of the center is 2, and we know the y-coordinate is 1 (since the center lies on the line y=1). Therefore, the center of the circle is (2,1).Matching Center with Answer Choices: Match the found center with the given answer choices. The center (2,1) matches with answer choice (A).

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