We know that arg of a complex no. is zero iff it lies on the +ve side of real axis.So,Let z₁=x₁+iy₁ and z₂=x₂+iy₂ then

im(z₁z₂)=0 iff x₁y₂+x₂y₁=0 and

arg of z₁+z₂=(x₁+x₂)+i(y₁+y₂) is 0 iff x₁+x₂>0 and y₁+y₂=0.

So,(x₂-x₁)y₁=0 x₂=x₁ or y₁=0

If x₂=x₁=x,y₂=-y₁=-y then we have z₁=x+iy and z₂=x-iy which are conjugate to each other.

If y₁=0 then both z₁ and z₂ should be real no.s such that their sum is positive.

Is the answer only option (C)?