How can we cut the bonds with the family when it is our duty to take care of them?

Duty is different from attachment. Krishna is Narayana. Narayana is Shiva. Manmadha was the only son of Narayana. But when Manmadha committed a mistake, Narayana in the form of Shiva burnt His only son. Dhritharashtra had a hundred sons. He could not punish even one son (Duryodhana). This is the difference between a scholar and an ignorant person. An ignorant person does his duties with attachment to the bonds. A scholar also does his duties but without attachment to the bonds. All these family bonds are temporary and confined only to this birth. In the previous birth, you were not related and in the future birth you will not be related with them. Shankara has told that that which is temporary must be unreal (Yat Anityam Tat Kritakamhi Loke). The two actors were not father and son before the drama and will not be father and son after the drama. The bond is temporary and exists only during the drama. But if you analyze further, even during the drama they are not father and son. Therefore that which is temporary must be unreal in the past, present and future. That which is permanent is always real. If you realize this divine knowledge of Shankara you can cut all your worldly bonds. Shankara left His old mother for the sake of the Lord. He practiced whatever He preached.