"Where is the city you live in?"
Translation:Wo ist die Stadt, in der du lebst?
Both work here.
I'm wondering if Duo shouldn't put "Wo liegt die Stadt..." as the suggested translation ( = "where is the city geographically located?"). "Wo ist die Stadt..." is fine, but for some reason I'm imagining a person (A) who promised to show someone (B) that city (e.g. "you'll be able to see it if we climb that hill over there"), brings (B) to that place, but there's no city to be seen, so (B) asks, "So now where's that city you said you live in?"
They're often completely interchangeable.
"in Berlin wohnen" = to inhabit a house/apartment in Berlin (and occasionally wash one's clothes and dust the cupoards there)
"in Berlin leben" = to spend one's life in Berlin (and enjoy the theatres and restaurants there)
"Ich wohne in Berlin, aber ich arbeite in Potsdam."
"Der Künstler (artist) lebt und arbeitet in Berlin."
Personally, if I'm asked where I live, I'd prefer "Ich wohne in [Stadt]" over "Ich lebe...".
Good question. "Worin" is generally a formal word, but it is used in formal spoken language, e.g. "der Raum, worin wir uns befinden, ist 300 Jahre alt" ("the room we are in is 300 years old"), "ein Dokument, worin der Ablauf beschrieben wird" ("a document in which the procedure is described")...
"Die Stadt, worin du wohnst" somehow doesn't feel very natural to me. I'm not sure if it's just me; if I google "Stadt, worin", I only get old-timey texts, but that doesn't mean it's wrong today. "Die Stadt, in der..." is way more common, though.
"Where is the city you live in?" I am assuming that the German translations use good grammar, but I think that the English grammar in this Relative Pronoun section is often poor. For example we would have been taught at school that: "Where is the city you live in?" is incorrect and should be: "Where is the city in which you live?". A modern alternative might be "Where is the city that you live in?" if one allows a sentence to finish with 'in'.
This is an old bad myth; it seems to have started froma couple of cranky remarks by John Dryden, who in his time was both immensely popular and greatly disliked -- but not a grammarian. Here is a link to a nice history of the myth, with an excellent discussion of proper usage. https://data.grammarbook.com/blog/definitions/dont-end-a-sentence-with-a-preposition-where-did-this-myth-come-from/