It is well-known that a circle can be uniquely defined by three points on its circumference, providing they are not collinear.
I have ascertained that the analogous number of points needed to uniquely define an ellipse is five, with the condition that each of those five points lies strictly outside the quadrilateral formed by the other four.
What I don’t know is how, given the planar coordinates of the five points, you could discover other information about the ellipse, such as axis lengths, orientation and position. I’m not at all sure there is a method that will work in the general case.
In some specific cases, it is possible to work out the position, orientation and axes of the ellipse, given the co-ordinates of five points.
For instance, given the points (1,0), (2,0), (0,1), (0,2) and (1,2) lying on an ellipse, find the length of the minor axis.