  In this paper, we use the properties of the self-polar triangle to not only
show a novel method for a basic point-line enumerative problem of conics, but
also present a series of closed-form solutions to the conics from all minimal
configurations of points and lines in general position. These closed-form
formulae may allow us to derive easily the algebraic and geometric conditions
which characterize when the obtained conic is real and non-degenerate, so we
propose a criterion for a non-degenerate real conic from each of all minimal
configurations. The correctness of our results is validated by some examples.
