y 6 4 v- 2'_\\ / - X 0 - -6 -4 -2 2- f- \\ I I',._ I~ '4 I '6 I Circle A (shown) is defined by the equation (x+2)^2 +y^2 = 9. Circle B (not shown) 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. Which equation defines circle B? (x+2)^2 +(y +6)^2 = (4)(9)
","rationale_html":"Choice A is correct. According to the graph, the center of circle A has coordinates -2,0 ^ h, 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 -2,0 6 units below the center of circle A. The point that’s 6 units below ^ h has the -2,0 same x-coordinate as ^ h and has a y-coordinate that is 6 less than the -2,0 y-coordinate of ^ h. Therefore, the coordinates of the center of circle B are -2,0-6 -2,-6 ^ h, or ^ h. Since the radius of circle B is 2 times the radius of 2 3 circle A, the radius of circle B is ^ h^ h. A circle in the xy-plane can be defined - 2+ - 2= 2 by an equation of the form ^x hh ^y kh r , where the coordinates of the center of the circle are ^h,kh and the radius of the circle is r. - - 2 3 Substituting 2 for h, 6 for k, and ^ h^ h for r in this equation yields - -2 2+ - -6 2= 2 3 2 ^x ^ hh ^y ^ hh ^^ h^ hh , which is equivalent to +2 2+ +6 2= 2 2 3 2 +2 2+ +6 2= 4 9 ^x h ^y h ^ h^ h , or ^x h ^y h ^ h^ h. Therefore, the +2 2+ +6 2= 4 9 equation ^x h ^y h ^ h^ h 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.
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