In the straight line case, in general each line will pass through at most two points if no three are collinear, and each point needs at least two lines, so you need as many lines as points (the fraction mentioned below is $\frac{2}{2}$) if there are at least three points. If there are two points, you need three lines, if one point then two lines.
Typically if there are more than three points and no three are collinear then there will be extra intersections.
For conic sections you get something similar: each conic section will pass through up to five points and each point needs two conic sections, so the number of conic sections must be at least $\frac{2}{5}$ of the number of points (rounded up), with two conic sections needed if you have one or two points, and three needed if you have five points. For circles the pattern is similar, with the fraction becoming $\frac{2}{3}$, with two circles needed for one point and three circles need for three points; with circles you can avoid extra intersections for example by using extra circles which touch tangentially rather than intersecting twice.