I would want to know WHY the 17 year old is looking to live with a friend or family member before I made judgements about the mother and her hypothetical approval.
If the 17 year old were in the last year of high school and the rest of the family were moving out of state, I don't think it would be unreasonable to allow her to live with her grandmother or her best friend's family to finish the school year with her class, for example.