Question 3449: z is an element of C such that [(z)/(z-i)] is real.How can I show that z is imaginary.
Answer by khwang(438)   (Show Source):
You can put this solution on YOUR website! Let z = a+bi where a,b are real.
If z/(z-i) = c for some real c.
then a + b i = c(z-i)
or a + b i = c(a+(b-1)i)
or a + b i = ca + c(b-1)i
So, a=ca and b = c(b-1)
a=ca implies a(c-1) = 0 Hence, a=0 or c =1
But, if c = 1, then b = c(b-1) implies b = b-1 impossible.
Hence, we see that a = 0, z = bi
If b = 0, then z = 0 and c = 0
and so z is imaginary or zero.
Kenny
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