The correct answer to the question is option 2: Colin Mackenzie. Colin Mackenzie was a British surveyor and engineer who played a significant role in bringing the ruins of Vijayanagara, also known as Hampi, to public attention in the year 1800.

Option 1: James Prinsep was not the surveyor and engineer responsible for bringing the ruins of Vijayanagara to public light. James Prinsep was an English scholar and antiquary known for his contributions to the decipherment of ancient scripts, particularly the Brahmi script used in ancient India.

Option 3: James Rennell was not involved in bringing the ruins of Vijayanagara to public light. James Rennell was a British geographer and cartographer known for his mapping of India and contributions to understanding ocean currents.

Option 4: Charles Metcalfe was not the surveyor and engineer who discovered the ruins of Vijayanagara. Charles Metcalfe was a British colonial administrator who served in various positions in the British East India Company in India.

In summary, Colin Mackenzie was the British surveyor and engineer who is credited with uncovering the ruins of Vijayanagara in 1800.