Let OA.OB=1  and  let O, A, B, be three collinear points. if O and B represent the complex numbers 0 and z, then A represents:

Given that OA.OB=1 ⇒|z1−0|.|z−0|=1⇒|z1|=1|z|....................(1) Also, amp(z1−0z−0)=0  or  π⇒amp(z1z)=0  or  π ⇒amp(z1)−amp(z)=0  or  π ⇒ampz1=amp z or amp(z)+π Let θ  be the amp of z1 ,then z1 =|z1|(cosθ+isinθ)                                                            =1|z|(cosθ+isinθ)             [using eq(1)]                                                           =1|z|2|z1|(cosθ+isinθ)=1|z|2(±z).                                                            =±zzz¯=±1z            ∴z1=1z  or  −1z