let
z = x+iy
k = a+ib where a,x,y and b are real and y and b =/= 0.
z+k, we have x+a+i(y+b), but since z+k is real, y+b = 0 and .'. y = -b or b = -y
zk = ax + ixb + aiy - yb
that is, xb + ay = 0 as zk is real.
from y = -b
xb - ab = 0
b(x-a) = 0
x-a = 0
x = a
Gathering it all...