The Northern Circars, comprising coastal Andhra districts, came under direct British control after the Battle of:

The correct answer is: b) Buxar (1764)

The Northern Circars were a group of provinces in the Indian subcontinent that were under the control of the Mughal Empire. In 1764, the British East India Company defeated the Mughals at the Battle of Buxar. This victory gave the British control of the Northern Circars. The British ruled the Northern Circars until 1947, when they were transferred to the newly independent Dominion of India.

The other options are incorrect because:

• Option a) is incorrect because the Battle of Plassey was fought in 1757, and the Northern Circars did not come under direct British control until 1764.
• Option c) is incorrect because the Battle of Buxar was fought in 1764, and the Northern Circars came under direct British control after that battle.
• Option d) is incorrect because the Third Anglo-Mysore War was fought in 1790-92, and the Northern Circars had already come under direct British control by that time.