Is zakat obligatory on a house that has been rented out for 3 years, given that its price was paid off only 6 months ago, and the rental income is used to cover family expenses and pay off the remaining installments on the house?

Zakat is not obligatory on this house because it is prepared for personal use. Zakat is only obligatory on its rent if it reaches the nisab (threshold) and a hawl (full lunar year) passes over it. However, if it is spent before a hawl passes over it, then there is no Zakat due on it.