Generalized pencils of conics derived from cubics

Abstract

Given a cubic K. Then for each point P there is a conic CP associated to P. The conic CP is called the polar conic of K with respect to the pole P. We investigate the situation when two conics C0 and C1 are polar conics of K with respect to some poles P0 and P1, respectively. First we show that for any point Q on the line P0P1, the polar conic CQ of K with respect to Q belongs to the linear pencil of C0 and C1, and vice versa. Then we show that two given conics C0 and C1 can always be considered as polar conics of some cubic K with respect to some poles P0 and P1. Moreover, we show that P1 is determined by P0, but neither the cubic nor the point P0 is determined by the conics C0 and C1.