Geometry of weak contact conics to irreducible quartics with 2 nodes and 1 cusp via rational elliptic surfaces and Zariski pairs

Let $\mathcal{Q}$ be an irreducible quartic with two nodes and one cusp as its singularities and let $\mathcal{C}$ be a conic such that the intersection multiplicity at each point of $\mathcal{C}\cap \mathcal{Q}$ is even and $\mathcal{C}\cap \mathcal{Q}$ contain at least one smooth point $z_o$ of $\mathcal{Q}$. In this paper, for every $\mathcal{Q}$ we find all possible conics $\mathcal{C}$ as above via studying geometry of $\mathcal{C}$ and $\mathcal{Q}$ through that of integral sections of a rational elliptic surface which canonically arises from $\mathcal{Q}$ and $z_o\in \mathcal{C}\cap \mathcal{Q}$. As an application, we construct Zariski pairs of degree 7 and degree 8, whose irreducible components consist of $\mathcal{Q}$, $\mathcal{C}$ and line passing through two of the singular points of $\mathcal{Q}$ .