"No es su hogar."
Translation:It is not her home.
In English we have lots of different words for the same thing and sometimes the same word can have many meanings. I imagine the same is true in Spanish, and if we want to be fluent we need to learn them all. Only by lots of real conversation with natives and reading will we slowly learn the subtle distinctions. I looked up "hearth" in a english-spanish dictionary and "hogar" was given first. So I assume we will know from context whether a speaker is referring to the hearth or to the home. I'm just guessing, though.
One of the possible answers given is "It is not her home." (You did put it in the negative, I assume.)
I got this question as a phrase to translate into English; perhaps Duolingo hasn't got a full set of possible answers in some of the other forms it offers the phrase?