In this note we study the limiting behaviour of the symbolic generic initial system {gin(I (m) )} of an ideal I ⊆ K[x, y, z] corresponding to an arrangement of r points of P 2 lying on an irreducible conic. In particular, we show that the limiting shape of this system is the subset of R 2 ≥0 such consisting of all points above the line through (min{ r 2 , 2}, 0) and (0, max{ r 2 , 2}).