Have you missed the rak'ah if you caught the imam standing in the first rak'ah before his bowing, then the imam bowed and rose before you were able to bow?

If you catch the Imam standing before he bows, you have caught the rak'ah. As for your lagging behind the Imam by a full pillar—which is bowing—it does not invalidate your prayer, especially if it was due to an excuse. If the congregant performs what he was preceded in and catches up with his Imam, there is nothing upon him. And if the lagging was without an excuse, then the prayer is not invalidated according to some scholars.