Given a general plane curve Y of degree d, we compute the number nd of irreducible plane conics that are five-fold tangent to Y. This problem has been studied before by Vainsencher using classical methods, but it could not be solved because the calculations produced too many non-enumerative correction terms that could not be analyzed. In our current approach, we express the number nd in terms of relative Gromov–Witten invariants that can then be directly computed. As an application, we consider the K3 surface given as the double cover of $\mathbb{P}^2$ branched along a sextic curve. We compute the number of rational curves in this K3 surface in the homology class that is the pull-back of conics in $\mathbb{P}^2$, and compare this number with the corresponding Yau–Zaslow K3 invariant. This gives an example of such a K3 invariant for a non-primitive homology class.