Is it permissible for me to pay zakat to my mother and brother, knowing that I provide for their food, drink, and clothing, with the exception of housing which my sister provides? And is it permissible to pay zakat to my brother for private school tuition fees?

A poor mother's maintenance is obligatory upon her capable children, so it is not permissible to give her zakat (definition). It is permissible to give zakat to a poor brother who is unable to earn, even if he is being supported voluntarily. It is also permissible to give him zakat to cover study expenses if he is diligent and engaged in acquiring knowledge that Muslims need, and he finds it difficult to combine seeking knowledge and earning, because the benefit of his knowledge will return to the Muslims.