The center of the circle is the point (negative 2 comma 0). Clockwise from top, the circle passes through the

Question

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?

Accepted Answer

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&rsquo;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 = r 2 , 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&rsquo;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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