2. (1996 National High School Competition Question) The figure formed by connecting the common points of the circle $x^{2}+(y-1)^{2}=1$ and the ellipse $9 x^{2}+(y+1)^{2}=9$ is ( ).
A. Line segment
B. Scalene triangle
C. Equilateral triangle
D. Quadrilateral
2. C. The coordinates of the intersection point $A(x, y)$ satisfy the equation $x^{2}+\frac{(y+1)^{2}}{9}=x^{2}+(y-1)^{2}$, solving which yields $y=2$ or $\frac{1}{2}$. Correspondingly, we find $x=0, x= \pm \frac{\sqrt{3}}{2}$, thus there are three intersection points $A_{1}(0,2), A_{2}\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right), A_{3}\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$. It is easy to see that $\left|A_{1} A_{2}\right|=\left|A_{1} A_{3}\right|=$ $\left|A_{2} A_{3}\right|=\sqrt{3}$, hence the resulting figure is an equilateral triangle, so the answer is $C$.