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Geometry and Trigonometry / Circles Difficulty: Medium

A circle in the xy-plane has the equation ${(x - 13)}^{2} + {(y - k)}^{2} = 64$ . Which of the following gives the center of the circle and its radius?

The center is at $left parenthesis 13 comma k right parenthesis$ and the radius is $8$ .

The center is at $left parenthesis k comma 13 right parenthesis$ and the radius is $8$ .

The center is at $left parenthesis k comma 13 right parenthesis$ and the radius is $64$ .

The center is at $left parenthesis 13 comma k right parenthesis$ and the radius is $64$ .

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Explanation

Choice A is correct. For a circle in the xy-plane that has the equation ${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$ , where $h$ , $k$ , and $r$ are constants, $(h, k)$ is the center of the circle and the positive value of $r$ is the radius of the circle. In the given equation, $h equals 13$ and $r squared equals 64$ . Taking the square root of each side of $r squared equals 64$ yields $r = \pm 8$ . Therefore, the center of the circle is at $left parenthesis 13 comma k right parenthesis$ and the radius is $8$ .

Choice B is incorrect. This gives the center and radius of a circle with equation ${(x - k)}^{2} + {(y - 13)}^{2} = 64$ , not ${(x - 13)}^{2} + {(y - k)}^{2} = 64$ .

Choice C is incorrect. This gives the center and radius of a circle with equation ${(x - k)}^{2} + {(y - 13)}^{2} = 4,096$ , not ${(x - 13)}^{2} + {(y - k)}^{2} = 64$ .

Choice D is incorrect. This gives the center and radius of a circle with equation ${(x - 13)}^{2} + {(y - k)}^{2} = 4,096$ , not ${(x - 13)}^{2} + {(y - k)}^{2} = 64$ .