A circle in the xy-plane has the equation x^(2)+y^(2)-10 x+32 y+272=0". " Which of the following best describes the location of the center of the circle and the length of its radius? Choose 1 answer: (A) Center: (10,-32) Radius: 4sqrt17 (B) Center: (-10,32) Radius: 4sqrt17 (c) Center: (-5,16) Radius: 3 (D) Center: (5,-16) Radius: 3

Question

A circle in the  xy-plane has the equation  x^(2)+y^(2)-10 x+32 y+272=0". " Which of the following best describes the location of the center of the circle and the length of its radius? Choose 1 answer: (A) Center:  (10,-32) Radius:  4sqrt17 (B) Center:  (-10,32) Radius:  4sqrt17 (c) Center:  (-5,16) Radius: 3 (D) Center:  (5,-16) Radius: 3

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Start Completing the Square: Write the given equation of the circle and start completing the square for the x and y terms. The given equation is x^2 + y^2 - 10x + 32y + 272 = 0. To complete the square for the x terms, we need to find the value that makes x^2 - 10x into a perfect square trinomial. The value needed is (10/2)^2 = 25. Similarly, for the y terms, we need to find the value that makes y^2 + 32y into a perfect square trinomial. The value needed is (32/2)^2 = 256.Add Necessary Values: Add and subtract the necessary values to complete the square inside the equation. We add 25 to both sides for the x terms and 256 to both sides for the y terms, and subtract these values outside the completed squares to keep the equation balanced. The equation becomes x^2 - 10x + 25 + y^2 + 32y + 256 = 272 + 25 + 256.Simplify the Equation: Simplify the equation by combining like terms and writing the completed squares. The equation now is (x^2 - 10x + 25) + (y^2 + 32y + 256) = 553. This simplifies to (x - 5)^2 + (y + 16)^2 = 553.Compare to Standard Form: Compare the simplified equation to the 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. From our equation (x - 5)^2 + (y + 16)^2 = 553, we can see that the center (h, k) is (5, -16) and r^2 is 553.Calculate the Radius: Calculate the radius of the circle. To find the radius r, we take the square root of 553. The radius r is √553. However, 553 is not a perfect square, so we need to simplify the square root. 553 = 17 * 32 + 9 = 17 * 33. So, the radius r is √(17 * 33) = √17 * √33. Since √33 is not an integer, we cannot simplify further, and the radius remains √553.Identify Correct Answer: Identify the correct answer from the given options. The center of the circle is (5, -16), and the radius is √553. None of the given options match this radius, so there must be a mistake in the calculation of the radius.

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