Translation:Please open your mouth.
You are not wrong. But "your" is more likely when someone says this to you. (Think of the dentist's office.) Using "the" in English sounds like you are asked to open the mouth of something/someone else...
Was fiddling with google translate, and it for some reason in this translation, it preferred あいて to あけて [開く vs 開ける].
Duo's was the transitive version of the verb, which makes more sense to me, as it's "you" opening "your mouth", but I wanted to know if this is the correct interpretation or if there were other differences?