Can you get social security benefits for a person that does not live with you?

Ok i will do my best explaining this..My mother(73) adopted a child(14) and because of my mother age age she is getting benefits for both herself and the child...but the child became to much for my mother and one day someone just drops her off at my doorstep and i have been taking care of her for the past 3 years... i have 2 kids my self and my mother refuse to help take care of her and has sent nothing to help pay for any expenses..she also refuses to take her back....so what i want to know is if it is legal for her to get the benefits when she is not taking care of her??? Should i and can be getting the benefits since i have been taking care of her?? If i can get the benefits how do i go about getting them??? I do want to add that my mother live in a different state as me and the child so i can prove that the child has been here using school records