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from time t=0 onwards, A has a constant nonzero velocity with magnitude v and a direction perpendicular to AoBo, where Ao and Bo are the initial positions of A and B........

I think"perpendicular to AoBo" is correct not "perpendicular to AB"

So A moves with a constant velocity (same direction).

B moves towards A and its direction will keep changing more and more towards that of A. Their relative velocity will tend towards zero. Their separation will tend to a constant value.

You will need integral calculus to solve this

You will see a similar problem in the first few pages of Irodv "Physics Problems"