Step 1: Distinguish empire name from city name: 'Vijayanagara' names both the empire (founded 1336 by Harihara and Bukka) and its capital city — the capital shared the empire's own name.
Step 2: Locate it: The capital sat on the Tungabhadra river, at the site now called Hampi.
Final Answer: Vijayanagara, on the Tungabhadra river (modern Hampi).