Let
z=x+iy
(z−a)
=(x−a)+i(y)
Argument is 2π3
Hence
(x−a)2(x−a)2+y2=14
3(x−a)2=y2 ...(i)
z+a
=(x+a)+iy
Now argument is π6
Hence
y2(x+a)2+y2=14
3y2=(x+a)2
9(x−a)2=(x+a)2
(x+a)=±(3x−3a)
x+a=3x−3a
x=2a and x+a=−3x+3a
x=a2
Hence
3y2=(x+a)2
3y2=9a2
y=±√3a
z=a(2±√3i)...(n)
or
3y2=9a24
y=±√3a2
z=a(1±√3i2)
Hence
|z|=|a(1±√3i2)|=a
Now we consider z=a(1+√3i2)
arg(z)=π3
Hence, option 'D' is correct.