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Geometry and Trigonometry / Circles Difficulty: Hard

The equation $x^{2} + {(y - 1)}^{2} = 49$ represents circle A. Circle B is obtained by shifting circle A down $2$ units in the xy-plane. Which of the following equations represents circle B?

${(x - 2)}^{2} + {(y - 1)}^{2} = 49$

$x^{2} + {(y - 3)}^{2} = 49$

${(x + 2)}^{2} + {(y - 1)}^{2} = 49$

$x^{2} + {(y + 1)}^{2} = 49$

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Explanation

Choice D is correct. The graph in the xy-plane of an equation of the form ${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$ is a circle with center $(h, k)$ and a radius of length $r$ . It's given that circle A is represented by $x^{2} + {(y - 1)}^{2} = 49$ , which can be rewritten as $x^{2} + {(y - 1)}^{2} = 7^{2}$ . Therefore, circle A has center $left parenthesis 0 comma 1 right parenthesis$ and a radius of length $7$ . Shifting circle A down two units is a rigid vertical translation of circle A that does not change its size or shape. Since circle B is obtained by shifting circle A down two units, it follows that circle B has the same radius as circle A, and for each point $(x, y)$ on circle A, the point $left parenthesis x comma y minus 2 right parenthesis$ lies on circle B. Moreover, if $(h, k)$ is the center of circle A, then $left parenthesis h comma k minus 2 right parenthesis$ is the center of circle B. Therefore, circle B has a radius of $7$ and the center of circle B is $left parenthesis 0 comma 1 minus 2 right parenthesis$ , or $left parenthesis 0 comma negative 1 right parenthesis$ . Thus, circle B can be represented by the equation $x^{2} + {(y + 1)}^{2} = 7^{2}$ , or $x^{2} + {(y + 1)}^{2} = 49$ .

Choice A is incorrect. This is the equation of a circle obtained by shifting circle A right $2$ units.

Choice B is incorrect. This is the equation of a circle obtained by shifting circle A up $2$ units.

Choice C is incorrect. This is the equation of a circle obtained by shifting circle A left $2$ units.