  The main goal of this note is to begin a systematic study on conic-line
arrangements in the complex projective plane. We show a de
Bruijn-Erd\H{o}s-type inequality and Hirzebruch-type inequality for a certain
class of conic-line arrangements having ordinary singularities. We will also
study, in detail, certain conic-line arrangements in the context of the
geography of log-surfaces and free divisors in the sense of Saito.
