The correct options are
A A can be written as x+iy where x,y∈R−{0}
B A is purely imaginary number
D A is both purely real as well as purely imaginary number
Given, z1¯¯¯z1=|z1|2=1
⇒¯¯¯z1=1z1
Likewise, ¯¯¯z2=1z2
We have A=z1+z21+z1z2
⇒¯¯¯¯A=¯¯¯¯¯z1+¯¯¯¯¯z21+¯¯¯¯¯z1⋅¯¯¯¯¯z2⇒¯¯¯¯A=1z1+1z21+1z1⋅1z2=z1+z21+z1z2
As ¯¯¯¯A=A, so A is a purely real number.