Choice C is correct. The equation of a circle in the xy-plane can be written as $(x - h)^{2} + (y - k)^{2} = r^{2}$ , where the center of the circle is $(h, k)$ and the radius of the circle is $r$ . It's given that the circle in the xy-plane is defined by the equation $(x - 1)^{2} + (y + 7)^{2} = 1$ . This equation can be written as $(x - 1)^{2} + (y - (- 7))^{2} = 1$ . For this equation, it follows that $h equals 1$ and $k = - 7$ . Therefore, the center of the circle in the xy-plane defined by the given equation is $left parenthesis 1 comma negative 7 right parenthesis$ .

Choice A is incorrect. This is the center of the circle in the xy-plane that is defined by the equation $(x + 1)^{2} + (y + 7)^{2} = 1$ , not $(x - 1)^{2} + (y + 7)^{2} = 1$ .

Choice B is incorrect. This is the center of the circle in the xy-plane that is defined by the equation $(x + 1)^{2} + (y - 7)^{2} = 1$ , not $(x - 1)^{2} + (y + 7)^{2} = 1$ .

Choice D is incorrect. This is the center of the circle in the xy-plane that is defined by the equation $(x - 1)^{2} + (y - 7)^{2} = 1$ , not $(x - 1)^{2} + (y + 7)^{2} = 1$ .