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Geometry and Trigonometry Difficulty: Hard

$x^{2} + 14 x + y^{2} = 6 y + 109$

In the xy-plane, the graph of the given equation is a circle. What is the length of the circle's radius?

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Explanation

Choice C is correct. It's given that in the xy-plane, the graph of the given equation is a circle. The equation of a circle in the xy-plane can be written in the form ${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$ , where $(h, k)$ is the center of the circle and $r$ is the length of the circle's radius. Subtracting $6 y$ from both sides of the equation $x^{2} + 14 x + y^{2} = 6 y + 109$ yields $x^{2} + 14 x + y^{2} - 6 y = 109$ . By completing the square, this equation can be rewritten as $(x^{2} + 14 x + {(\frac{14}{2})}^{2}) + (y^{2} - 6 y + {(\frac{- 6}{2})}^{2}) = 109 + {(\frac{14}{2})}^{2} + {(\frac{- 6}{2})}^{2}$ . This equation can be rewritten as $(x^{2} + 14 x + 49) + (y^{2} - 6 y + 9) = 109 + 49 + 9$ , or ${(x + 7)}^{2} + {(y - 3)}^{2} = 167$ . Therefore, $r squared equals 167$ . Taking the square root of both sides of this equation yields $r equals StartRoot 167 EndRoot$ and $r = - \sqrt{167}$ . Since $r$ is the length of the circle's radius, $r$ must be positive. Therefore, the length of the circle's radius is $StartRoot 167 EndRoot$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.