Circle
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A sphere
Parabola - apex
One.
Yes, you can draw a circle through the four points of a trapezoid, because the four points of the trapezoid can be equidistant from one point, making that distance the radius.
You can't. It is impossible to have five equidistant points on a sphere, with the exception of trivial cases (i.e. where the radius is 0 or when one or more points are equal).