Although it was designated a royal centre, the south-western part of…

The correct answer is Option (1) → Rulers sought to legitimise their authority through association with the divinities housed there

Option (1) Rulers sought to legitimise their authority through association with the divinities housed there — Correct. Rulers were trying to establish and legitimise their authority. Linking themselves with the divinities housed in the shrines did this, so they patronised many temples.

Option (2) Different communities living near the palaces each supported a cult of their own there — Incorrect. Many cults backed by different communities explain the small shrines of the urban core. Temples in the royal centre reflect royal patronage.

Option (3) Rulers needed the temple tanks there as the main water supply for the urban core — Incorrect. Wells, rainwater tanks and temple tanks served ordinary town dwellers in the urban core. Water does not explain over 60 temples in the royal centre.

Option (4) Temples there served as granaries to feed the defenders during long sieges — Incorrect. To face sieges, Vijayanagara rulers protected the agricultural belt inside the walls, a costlier plan than granaries. Temples were not built as granaries.