All perfect praise be to Allah, The Lord of the Worlds. I testify that none is worthy of worship except Allah and that Muhammad is His slave and messenger.
The scholarly opinion we adopt in Islamweb is that it is obligatory on the person led in prayer to recite Al-Faatihah. If the Imam performs Rukoo‘ (bowing) before those led in prayer finish reciting Al-Faatihah, they should complete the recitation and then follow the Imam. There is no harm if the Imam finishes the Rukoo‘ before they finish the recitation; rather, they should complete the recitation and then follow the Imam in the rest of the prayer. Thus, you come to know that there is nothing wrong in completing the recitation of Al-Faatihah after the Imam had performed Rukoo‘; you should then perform Rukoo‘ after finishing the recitation.
Your prayer is valid whether you managed to catch up with the Imam when he was still performing the Rukoo‘ or you did not catch up with him. This is the view of the Shaafi‘i School.
Allah Knows best.