Rectifiable pencils of conics

We describe analytic pencils of conics passing through the origin in $\mathbb C^2$ that can be mapped to straight lines locally near the origin by an analytic diffeomorphism. Under a minor non-degeneracy assumption, we prove that in a pencil with this property, almost all conics have 3 points of tangency with the same algebraic curve of class 3 (i.e. a curve projectively dual to a cubic).