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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Rewriting the equation: To find the center and radius of the circle, we need to rewrite the equation in standard form, which is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius.Grouping the terms: First, we group the x terms and the y terms together: (x^2 - 10x) + (y^2 + 32y) = -272.Completing the square for x terms: Next, we complete the square for the x terms and the y terms. To complete the square, we take half of the coefficient of x (which is -10) and square it, adding and subtracting this value inside the parentheses. We do the same for y with the coefficient of 32.Completing the square for y terms: For the x terms: (x^2 - 10x + (10/2)^2) - (10/2)^2 = (x^2 - 10x + 25) - 25. For the y terms: (y^2 + 32y + (32/2)^2) - (32/2)^2 = (y^2 + 32y + 256) - 256.Adding completed squares to both sides: Now we add these completed squares to both sides of the equation: (x^2 - 10x + 25) + (y^2 + 32y + 256) = -272 + 25 + 256.Simplifying the equation: Simplify the equation: (x - 5)^2 + (y + 16)^2 = 9.Identifying the center and radius: Now we can see that the equation is in standard form, with the center of the circle at (h, k) = (5, -16) and the radius r = √9 = 3.Correct answer: Comparing with the given options, the correct answer is (D) Center: (5, -16) Radius: 3.

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