1. Let $P_{0}\left(x_{0}, y_{0}\right)$ be any point on the circle $x^{2}+(y-1)^{2}=1$, and we want the inequality $x_{0}+y_{0}+c \geqslant 0$ to always hold. Then the range of values for $c$ is ( ).
(A) $[0,+\infty$ )
(B) $[\sqrt{2}-1, \cdots-\infty$ )
(C) $(-\infty, \sqrt{2}+1]$
(D) $(1 \cdots \sqrt{2},+\infty)$
-1 . B.
Circle $x^{2}+(y-1)^{2}=1$ should be above the line $x+y+c=0$, meaning the line $x+y+c=0$ should be below or tangent to the circle, or separate from it. As shown in the figure, $A(0,1-\sqrt{2})$, hence $c \geqslant \sqrt{2}-1$.