Multiple Conics Passing Through Four Points
This applet demonstrates that, in general, two (distinct, nondegenerate) conic sections intersect in four points. Therefore, four points do not uniquely determine a conic section. Two different conic sections are drawn through the points A, B, C, and D. One of these conics passes through P1; the other passes through P2.
In the applet, you may click and drag the points A, B, C, and D; conics will be drawn passing through these points as described above. (The points P1 and P2 are stationary.)
Bradley and Stemkoski, Created with GeoGebra