If `z=x+i y and w=(1-i z)/(z-i)` , show that `|w|=1 z` is purely real.

Explanation

We have
`|w| = 1`
`rArr |(1-iz)/(z-i)| = 1 or (|1-iz|)/(|z-i|)=1`
`or | 1 - iz | = |z-i|`
or `|1-i(x+iy)|=|x+iy -i|,` where z = x + iy
`or |1+y-ix|=|x +i(y-1)|`
`or sqrt((1+y)^(2) +(-x)^(2))=sqrt(x^(2) + (y-1)^(2))`
`or (1+y)^(2) + x^(2) =x^(2) +(y-1)^(2)`
` or y = 0`
`rArr z = x +i0 = x`, which is purely real