A teacher's job has complexities. A private studio teacher can have several faces to the student. I am interested in the person finding their way in the field. I am also interested in the person as a human. But I am not a trained psychologist like my wife, Dr. Carol Viera, who is able to deal with both the musical and human aspects at many different levels. However, I do try to deal with certain basic things, if I see it is within my ability and not go beyond that. It can be a fine line! But, if there is real care and sensitivity involved, the right course of action has a better chance of appearing.
I think it is important that the trust is key in the relationship. It is also a nice added touch if student and teacher happen to like each other as people. That does not mean students and teachers have to be friends as such but can have a warm interaction with respect towards one another.
Depending how big the age gap gets, it can feel more like parent-offspring relations in a way. When I started teaching at the college level, the BIGGEST gap between a student and myself was 2 years or so. That is basically the same age and it was easy to become actual friends and hang out together. Now with incoming freshman it is 38 years. BIG DIFFERENCE! That does not make all the relationships the same because you are dealing with unique people and chemistry. When the music really gets happening or there is a strong connection and flow between student and teacher in the territory being made, any age gap disappears into the shared experience. All become ageless.