1. Connecting the common points of the circle $x^{2}+(y-1)^{2}=1$ and the ellipse $9 x^{2}+(y+1)^{2}=9$ forms the figure of
(A) a line segment
(B) a scalene triangle
(C) an equilateral triangle
(D) a quadrilateral
(C)
I. Multiple Choice Question
1.【Analysis and Solution】Solving the equation $x^{2}+\frac{(y+1)^{2}}{9}=x^{2}+(y-1)^{2}$ yields $y=2$ or $y=\frac{1}{2}$, then substituting these values into the circle equation gives $x=0$ or $x= \pm \frac{\sqrt{3}}{2}$. Thus, we obtain three intersection points of the two figures: $A_{1}(0,2), A_{2}$ $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right), A_{3}\left(\frac{-\sqrt{3}}{2}, \frac{1}{2}\right)$. Considering the options include scalene triangles and equilateral triangles, we will determine whether these three intersection points form an equilateral triangle.
It is easy to see that $\left|A_{1} A_{2}\right|=\left|A_{1} A_{3}\right|=\left|A_{2} A_{3}\right|$, so the resulting figure is an equilateral triangle.