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)

Bytelearn · Accepted Answer

Identify Formula: Identify the formula to find the distance between two points in the xy-plane. The distance formula is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$, where $d$ is the distance between the points $(x_1, y_1)$ and $(x_2, y_2)$.Use Distance Formula: Given that the point (2, -3) lies on the circle and the radius of the circle is 4, we can use the distance formula to check which of the given center options satisfies the condition that the distance from the center to the point (2, -3) is equal to 4.Check Option (A): First, check option (A) (2, 1) as the center. Using the distance formula with (x_1, y_1) = (2, 1) and (x_2, y_2) = (2, -3), we get $d = \sqrt{(2 - 2)^2 + (-3 - 1)^2} = \sqrt{0 + (-4)^2} = \sqrt{16} = 4$.Verify Center: Since the distance from the center (2, 1) to the point (2, -3) is exactly 4, which matches the given radius of the circle, option (A) (2, 1) could be the center of the circle.Final Conclusion: There's no need to check the other options because the question asks for which of the following could be the center, and we have found a valid option that satisfies the given conditions.

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