Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) was asked about the ruling on giving zakaat al-fitr to poor relatives. 

He replied:   

It is permissible to give zakaat al-fitr and zakaah of one’s wealth to relatives who are poor, and in fact giving it to relatives is better than giving it to strangers because giving it to relatives is both charity and upholding ties of kinship. But that is subject to the condition that giving it is not a means of protecting one’s wealth, i.e., if this poor person is one on whom the rich person is obliged to spend, in this case it is not permissible to meet his needs by giving zakaah, because if he does that he will have saved his own money by giving him zakaah, and this is not permissible. But if he is not obliged to spend on him, then he may give his zakaah to him, and giving zakaah to him is better than giving it to a stranger, because the Prophet (peace and blessings of Allaah be upon him) said: “Your charity given to a relative is both charity and upholding the ties of kinship.” End quote. 

Based on this, it is not permissible for you to give your zakaat al-fitr to your mother, rather you should spend on her from wealth other than your zakaah.