OG详解-OG7 数学2 Q25

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 = r² , where (h, k) is the center of the circle and r is the length of the circle’s radius. Subtracting 6y from both sides of the equation x² +14x+y² = 6y+109 yields x² +14x+y² -6y = 109.   By completing the square, this equation can be rewritten as (x² +14x+ (14​2​ )²) + (y² -6y+ (−6​2​ )²) = 109+ (14​2​)²+ (−6​2​)² . This equation can be rewritten as (x² +14x+49)+(y² -6y+9) = 109+49+9, or (x+7)² +(y-3)² = 167. Therefore, r² = 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.