My understanding is that adding the article to a possessive pronoun makes it function basically like a noun. Remember that via here is an adjective modifying pilko. It indicates that the ball is a ball possessed by you.
On the other hand, la via would behave like a noun. Since both these parts of speech are the same -yours- in English, you can't readily see the difference in English and so we have to use word order to indicate it:
So "Kiu pilko estas la via" would actually be "Yours is which ball?" Here yours is (loosely speaking) a noun. This is possible because Esperanto has flexible word order. If we wanted it to be in a more natural word order for English speakers, it would be "La via estas kiu pilko?"
See point 2
English bites us in the butt again. I would say that "Which ball is yours?" and "Yours is which ball?" both have the same meaning. Actually I was wondering if Duo made a mistake by not saying 'la via' since that's the way "yours" has been presented to us in other lessons. But, I do see your point. Thanks.
"Kiu pilko estas la via pilko" sounds wrong to me; I would say either "Kiu pilko estas via pilko" or "Kiu pilko estas la via". I guess I'm basing this, however, on Dutch intuition, which might be wrong in this case, because in Dutch you also wouldn't be able to say "Kiu pilko estas via" without the "la" (the Dutch equivalent thereof, I mean, obviously).
Then again, it's similar in English as well: you can't say "Which ball is yours ball?".