The correct answer is: a) Harihara I and Bukka Raya I.
Harihara I and Bukka Raya I were brothers who founded the Vijayanagara Empire in 1336. The empire was founded in response to the Islamic invasions of the Deccan Plateau. Harihara I was the first ruler of the empire, and he was succeeded by his brother Bukka Raya I. The Vijayanagara Empire reached its peak under the rule of Krishna Deva Raya in the 16th century. The empire was a major power in South Asia for over 200 years, and it played a significant role in the development of Indian culture.
Option b) is incorrect because Krishna Deva Raya was a ruler of the Vijayanagara Empire, not its founder. Option c) is incorrect because Alauddin Khalji was a Muslim ruler who invaded the Deccan Plateau in the 14th century. Option d) is incorrect because Shivaji was a Hindu ruler who founded the Maratha Empire in the 17th century.