The correct answer is option 3, Krishna. The Krishna river originates in the state of Maharashtra and flows through the states of Karnataka and Andhra Pradesh before finally meeting the Bay of Bengal in the Andhra coast.

Let`s examine the other options for better understanding:

Option 1: Mahanadi. The Mahanadi river is not the correct answer because it does not flow through the states of Karnataka and Andhra Pradesh as stated in the question. Instead, it flows through the states of Chhattisgarh and Odisha.

Option 2: Cauvery. The Cauvery river is also not the correct answer as it does not originate in Maharashtra. Instead, it originates in the state of Karnataka and flows through Tamil Nadu before finally emptying into the Bay of Bengal.

Option 4: Godavari. The Godavari river is not the correct answer either as it does not flow through Karnataka. It originates in Maharashtra and flows through Telangana and Andhra Pradesh before meeting the Bay of Bengal in the Andhra coast.

In conclusion, the correct answer is option 3, Krishna, as it originates in Maharashtra, flows through Karnataka and Andhra Pradesh, and meets the Bay of Bengal in the Andhra coast.