OG详解-OG6 数学1 Q23

Choice A is correct. According to the graph, the center of circle A has coordinates (-2, 0), and the radius of circle A is 3. It’s given that circle B is the result of shifting circle A down 6 units and increasing the radius so that the radius of circle B is 2 times the radius of circle A. It follows that the center of circle B is 6 units below the center of circle A. The point that’s 6 units below (-2, 0) has the same x-coordinate as (-2, 0) and has a y-coordinate that is 6 less than the y-coordinate of (-2, 0). Therefore, the coordinates of the center of circle B are (-2, 0-6), or (-2, -6). Since the radius of circle B is 2 times the radius of circle A, the radius of circle B is (2)(3). A circle in the xy-plane can be defined by an equation of the form (x-h)2 +(y-k)2 = r² , where the coordinates of the center of the circle are (h, k) and the radius of the circle is r.
Substituting -2 for h, -6 for k, and (2)(3) for r in this equation yields (x-(-2))² +(y-(-6))² = ((2)(3))², which is equivalent to (x+2)² +(y+6)² = (2)²(3)² , or (x+2)² +(y+6)² = (4)(9). Therefore, the equation (x+2)² +(y+6)² = (4)(9) defines circle B. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect. This equation defines a circle that’s the result of shifting circle A up, not down, by 6 units and increasing the radius. Choice D is incorrect and may result from conceptual or calculation errors.