2. The curve represented by the equation $|y|=1+\sqrt{2 x-x^{2}}$ is ( ).
(A) a circle
(B) two semicircles
(C) an ellipse
(D) none of the above conclusions is correct
2. B.

From the given equations, we have
$$
y=1+\sqrt{2 x-x^{2}} \text { and } y=-1-\sqrt{2 x-x^{2}},
$$

which represent the upper semicircle with center at $(1,1)$ and radius 1, lying between the points $(0,1)$ and $(2,1)$, and the lower semicircle with center at $(1,-1)$ and radius 1, lying between the points $(0,-1)$ and $(2,-1)$, respectively.