Answer

In principle, after 24 hours it is generally accepted that the taste absorbed in the pot has become spoiled. Therefore, if forbidden food (or meat in a dairy pot, or vice versa) was cooked in it, the food is permitted to be eaten after the fact. Ideally, however, the Sages prohibited cooking in such a pot, lest one come to cook in it before 24 hours have passed. But if one cooked unintentionally, the food does not become forbidden (the decree did not apply to that case). That is true for all prohibitions. As for chametz on Passover, Sephardim treat it like any other prohibition, but Ashkenazim are more stringent: even the slightest amount of chametz renders it forbidden, even if its taste is spoiled (Orach Chayim 447:10). It seems to me that there is room to argue that in the case of chametz this is forbidden by the basic law itself (albeit rabbinically, since the rule that even the slightest amount of chametz is prohibited is entirely rabbinic), and not because of the decree mentioned above. Your question dealt with a case where others did the cooking, and in my opinion that is equivalent to your own intentional action, and it is forbidden to eat it, since you know that this is what they are doing and they are doing it for you as well. It seems to me that your family should understand and take into account that you observe kosher for Passover, and find a suitable arrangement. Happy holiday, Michi