A

Step 1: The speed of the man in still water is 3 m/s, and the river flows with a speed of 1 m/s.
Step 2: Let \( \theta \) be the angle at which the man swims with respect to the current. The velocity of the man can be decomposed into two components: one along the bank and one across the bank.
\( U_{man} = 3 \cos(\theta) \) (along the bank) and \( V_{man} = 3 \sin(\theta) \) (across the bank).
Step 3: The resultant velocity along the x-axis (bank) is given by:
\( U_{resultant} = U_{river} + U_{man} = 1 + 3\cos(\theta) \)
Step 4: The distance covered in the y-direction (across the river) in 1 second is:
\( y = 3\sin(\theta) \)
Step 5: The x-coordinate after 1 second is:
\( x = 1 + 3\cos(\theta) \)
Step 6: Using trigonometric identities, we have:
\( \sin^2(\theta) + \cos^2(\theta) = 1 \).
Step 7: Substitute \( \cos(\theta) = \frac{x - 1}{3} \) into the identity.
Step 8: Thus, \( y = 3\sqrt{1 - (\frac{x - 1}{3})^2} \).
Step 9: The equation of the locus can be simplified to:
\( y = \sqrt{9 - (x - 1)^2} \).
Therefore, the correct answer is A.