Now to tackle the general problem of giving a conic section from five elements.
I take the following approach:
I have two starting points from which the conic section can be given: from five tangents, we can draw it directly. From five points, we can solve the system of linear equations.
I will establish an analytic conversion between points and tangents, so that the two can be treated as the same case. Then I will tackle mixed conditions: [m points , n tangents], where there are always a total of five elements. For each mixed case, I will seek a suitable way (matrix transform, projection) to bring the number of known points ... or tangents ... to five, reducing the problem to one I can already solve.
Onward