I'm not sure if the police would get involved unless your father is actively violent towards you or your mom. But if he is violent, you should certainly go to the police.
If your dad is so concerned about keeping his image good in society, perhaps your mom (or your grandparents) could have a talk with him about improving his behavior at home. If he wants to be viewed as a nice and respectable man, he should behave himself at home. The truth has a way of catching up with us all, and one day people will know what he's really like.