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Geometry and Trigonometry / Circles Difficulty: Hard

${(x + 4)}^{2} + {(y - 19)}^{2} = 121$

The graph of the given equation is a circle in the xy-plane. The point $(a, b)$ lies on the circle. Which of the following is a possible value for $a$ ?

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Explanation

Choice B is correct. An equation of the form ${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$ , where $h$ , $k$ , and $r$ are constants, represents a circle in the xy-plane with center $(h, k)$ and radius $r$ . Therefore, the circle represented by the given equation has center $left parenthesis negative 4 comma 19 right parenthesis$ and radius $11$ . Since the center of the circle has an x-coordinate of $negative 4$ and the radius of the circle is $11$ , the least possible x-coordinate for any point on the circle is $- 4 - 11$ , or $negative 15$ . Similarly, the greatest possible x-coordinate for any point on the circle is $- 4 + 11$ , or $7$ . Therefore, if the point $(a, b)$ lies on the circle, it must be true that $- 15 \leq a \leq 7$ . Of the given choices, only $negative 14$ satisfies this inequality.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.