Question 539691
If z+w is real then *[tex \LARGE Im(z+w) = 0 \Rightarrow b+y = 0] (here, Im(z) stands for the imaginary part of z), so b = -y. Also, if zw is real, then


*[tex \LARGE Im((a+bi)(x+yi)) = 0]. We can replace y with -b.


*[tex \LARGE Im((a+bi)(x-bi)) = 0 \Rightarrow -abi + bxi = 0 \Rightarrow a = x]


Since a = x and y = -b, z and w are conjugates of each other.