What should be the ratio of velocities of proton to that of an alpha particle so that they trace a path of same radius in a perpendicular magnetic field ?

1 : 1

m_α = (4) m_p 

q_α = 4 q_p                       

r_α = (4 m_p) v_α / [ 4 q_p B]  = ( m_p) v_α / [q_p B]      

r_p = (m_p) v_p / [q_p B]          

r_α : r_p  = 1 : 1 

∴  v_p : v_α = 1 : 1