The correct option is C √5−12
az2+z+1=0 …(1)
Taking conjugate of both sides,
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯az2+z+1=¯0
⇒¯a(¯z)2+¯z+¯1=0
⇒¯az2−z+1=0 …(2)
(since ¯z=−z as z is purely imaginary)
Eliminating z from equations (1) and (2), we get
(¯a−a)2+2(a+¯a)=0 …(3)
Let,
a=cosθ+isinθ (∵|a|=1)
From equation (3) we get,
(−2isinθ)2+2(2cosθ)=0
⇒4cos2θ+4cosθ−4=0
⇒cos2θ+cosθ−1=0
⇒cosθ=−1±√1+42
∵θ=arg(a)
∴cos(arg(a))=√5−12