That position in a throne room does not have any name of its own (at least not in common use), but is instead often called the "foot of the throne" which implies inferior position.
The peasant bowed at the foot of the throne as he plead innocence.
Slightly less inferior (depending on context) is "before the throne", often used in the context of someone who has an audience (see definition 2) with the monarch on the throne.
The guards led the young man before the throne, where the king invited him to speak.
This use of "before" also sometimes is used to suggest the person is being judged in some way, as in the idiomatic expression "before the court" to talk about various judicial procedures that take place before a judge.
Note that the actual protocols of when and how to bow before a king vary considerably between different countries, cultures, and times. In some of the most formal, petitioners before the monarch would be expected to bow multiple times, in different parts of the room, before speaking. Because of this it is difficult to suggest that there is only one part of the room where people would bow.