x 2 + 14 x + y 2 = 6 y + 109 In the xy -plane, the graph of

Question

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?

Accepted Answer

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 + 14 2 2 + y 2 - 6 y + - 6 2 2 = 109 + 14 2 2 + - 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 2 = 167 . Taking the square root of both sides of this equation yields r = 167 and r = - 167 . Since r is the length of the circle's radius, r must be positive. Therefore, the length of the circle's radius is 167 . 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.

Question x-2-14