In Young's double slits experiment fringe width is β. Intensity at a point on the screen does not change when one of the slit is closed or opened, then the minimum distance of that point from the central maximum is

Intensity at any point on the screen is I= I0cos2δ2 where δ is the phase difference between the waves meeting at that point. When one of the slit is closed intensity at any point is due to one source i.e. I04given I04=I0cos2δ2 solving we get δ=2π3 i.e. corresponding path difference is λ3for λ path difference distance is β for λ3it is β3