So the duke is actually the brother who is supposed to be dead. They were twins but since his brother died he took his identity and became the head and is pretending to be the twins dad(the twins dad is the dead brother). Mc' mother and the duke were actually lovers but since the duke decided to use his brothers identity and pretending to be the one who died he couldn't be with her anymore and send her away to protect her from imperial family. But he didn't know she was pregnant. Before dying she send Mc to her grandpa who knew she was his granddaughter that's why he left her everything.
So maybe she is the child of that ruth guy