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Geometry and Trigonometry Difficulty: Hard

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?

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Explanation

Choice A is correct. According to the graph, the center of circle A has coordinates $left parenthesis negative 2 comma 0 right parenthesis$ , 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 $left parenthesis negative 2 comma 0 right parenthesis$ has the same x-coordinate as $left parenthesis negative 2 comma 0 right parenthesis$ and has a y-coordinate that is $6$ less than the y-coordinate of $left parenthesis negative 2 comma 0 right parenthesis$ . 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 $left parenthesis 2 right parenthesis left parenthesis 3 right parenthesis$ . 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 $negative 2$ for $h$ , $negative 6$ for $k$ , and $left parenthesis 2 right parenthesis left parenthesis 3 right parenthesis$ 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.