The correct answer is option 3, Vijayanagara. The Hazara Rama temple belongs to the Vijayanagara kingdom. Let`s analyze each option to better understand why they are or are not the correct answer.

Option 1: Avadh. This option is incorrect as the Hazara Rama temple does not belong to the Avadh kingdom. Avadh was a kingdom in present-day Uttar Pradesh, India, known for its cultural and architectural heritage, but the Hazara Rama temple is not associated with this kingdom.

Option 2: Travancore. This option is incorrect as well. Travancore was a kingdom in present-day Kerala, India. While Travancore has its own rich history and significant temples, the Hazara Rama temple does not belong to this kingdom.

Option 3: Vijayanagara. This is the correct answer. The Hazara Rama temple is indeed associated with the Vijayanagara kingdom. The temple is located in Hampi, Karnataka, India, and was built during the reign of the Vijayanagara rulers.

Option 4: Ahom. This option is incorrect. The Ahom kingdom was in the region of present-day Assam, India, but the Hazara Rama temple is not related to this kingdom