In this article, we consider Zariski N-plets whose irreducible components are an irreducible quartic and smooth conics. More precisely we give examples of Zariski N + 1-plets of degree 2N + 4 whose irreducible components are an irreducible quartic curve with 2 nodes and N smooth conics. We construct such examples as an application of our study of the geometry of bisections of certain rational elliptic surfaces. Moreover, by considering the case of N = 2 and combining with known results, a new Zariski 5-plet for reduced plane curves of degree 8 is given.
