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Geometry and Trigonometry / Circles Difficulty: Hard

The equation $x^{2} + {(y - 2)}^{2} = 36$ represents circle A. Circle B is obtained by shifting circle A down $4$ units in the xy-plane. Which of the following equations represents circle B?

$x^{2} + {(y + 2)}^{2} = 36$

$x^{2} + {(y - 6)}^{2} = 36$

${(x - 4)}^{2} + {(y - 2)}^{2} = 36$

${(x + 4)}^{2} + {(y - 2)}^{2} = 36$

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Explanation

Choice A is correct. The standard form of an equation of a circle in the xy-plane is ${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$ , where the coordinates of the center of the circle are $(h, k)$ and the length of the radius of the circle is $r$ . The equation of circle A, $x^{2} + {(y - 2)}^{2} = 36$ , can be rewritten as ${(x - 0)}^{2} + {(y - 2)}^{2} = 6^{2}$ . Therefore, the center of circle A is at $left parenthesis 0 comma 2 right parenthesis$ and the length of the radius of circle A is $6$ . If circle A is shifted down $4$ units, the y-coordinate of its center will decrease by $4$ ; the radius of the circle and the x-coordinate of its center will not change. Therefore, the center of circle B is at $left parenthesis 0 comma 2 minus 4 right parenthesis$ , or $left parenthesis 0 comma negative 2 right parenthesis$ , and its radius is $6$ . Substituting $0$ for $h$ , $negative 2$ for $k$ , and $6$ for $r$ in the equation ${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$ yields ${(x - 0)}^{2} + {(y - (- 2))}^{2} = {(6)}^{2}$ , or $x^{2} + {(y + 2)}^{2} = 36$ . Therefore, the equation $x^{2} + {(y + 2)}^{2} = 36$ represents circle B.

Choice B is incorrect. This equation represents a circle obtained by shifting circle A up, rather than down, $4$ units.

Choice C is incorrect. This equation represents a circle obtained by shifting circle A right, rather than down, $4$ units.

Choice D is incorrect. This equation represents a circle obtained by shifting circle A left, rather than down, $4$ units.