Putnam 1967 A6

Given real numbers $(a_i)$ and $(b_i)$ (for $i=1,2,3,4$ ) such that $a_1 b _2 \ne a_2 b_1 .$ Consider the set of all solutions $(x_1 ,x_2 ,x_3 , x_4)$ of the simultaneous equations $$a_1 x_1 +a _2 x_2 +a_3 x_3 +a_4 x_4 =0 \; \text{and}\; b_1 x_1 +b_2 x_2 +b_3 x_3 +b_4 x_4 =0$$ for which no $x_i$ is zero. Each such solution generates a $4$ -tuple of plus and minus signs (by considering the sign of $x_i$ ). [list=a] [] Determine, with proof, the maximum number of distinct $4$ -tuples possible. [] Investigate necessary and sufficient conditions on $(a_i)$ and $(b_i)$ such that the above maximum of distinct $4$ -tuples is attained.