The first post this series said that a conic section has five degrees of freedom, and that any theorem that claims to determine a conic by less than five numbers is using some additional implicit information.
The second post looked at Gibbs’ method which uses three observations, and a variation on the method uses just one observation. The post explains where the missing information is hiding.
The third post looked at Lambert’s theorem which determines a conic section from two observations and again explains where the additional information comes from.
You could also determine a conic by an assortment of points and tangent lines (physically, a set of observations and velocities). You could specify four points on the conic and a line it must be tangent to. Or you could specify three points and two tangents, etc. See, for example, here.
As a general rule, any five reasonable pieces of information about a conic section are enough to determine it uniquely. The tedious part is writing down the exceptions to the general rule. For example, you can’t have three of the points lie on a line.
You can’t always get by with saying “I’ve got n degrees of freedom and n pieces of information, so there must be a solution.” And yet remarkably often you can.