Right Answer:
Vivía
The word familia (family) is a Determined Collective Noun.
The Determined Collective Nouns are those that carry in their meaning the nature of their components. These nouns appear in singular with a plural meaning, in addition, they agree with the verb in singular when they constitute nominal groups according to the subject.
This is how the word Familia (Family: a group of people with kinship relationships, whether by marriage, blood or adoption) in this sentence is the 3rd person in singular.
Then, the spanish verb vivir (to live) in simple past is conjugated with the 3rd person singular as vivía (lived).
Therefore, the whole sentence is:
La familia Vargas vivía en una casa bonita.
(The Vargas family lived in a nice house)
The other options are incorrect because do not match with the noun of the sentence:
vivían (they lived) is the conjugation with the 3rd person plural
vivíamos (we lived) is the conjugation with the 1st person in plural
vivías (you lived) is the conjugation with the 2nd person in singular
