z=a+bi
|z|=1
a^2+b^2 =1 (1)
|z-1/2| =√3/2
(a-1/2)^2 +b^2 = 3/4 (2)
(1)-(2)
a-1/4 = 1/4
a= 1/2
b= √3/2 or -√3/2
z = 1/2 +(√3/2) i or 1/2 -(√3/2) i