The biased, distance-restricted n-in-a-row game for small p

The biased n-in-a-row game was shown to be a win for the first player for any n by J. Beck. To limit the advantage of picking more than one point per move he suggested a weak form of the game where the first player's p points for each move must be contained in a circle of radius r. For p=2, we give a tight bound for the maximum length of the line where the first player can force a win, answering an open problem posed by Csorba.