If `alpha = (z - i)//(z + i)` show that, when z lies above the real axis, `alpha`will lie within the unit circle which has centre at the origin. Find the locus of `alpha ` as z travels on the real axis form `-oo "to" + oo`

Explanation

From figure, it is clear that `|z-i| lt |z+i|` (as z lies above the real axis ). Hence,
`|alpha| = (|z-i|)/(|z+i|) lt 1`
Therefore, `alpha` lies withinn the unit circl which has centre at the e origin. Now if z is traveling on the real axis `Im (z) = 0, Re(z)` various from `-oo "to" +oo`. Let
`z=x +i0`
`rArr alpha = (x-i)/(x+i)`
`rArr |a| =(|x-i|)/(|x+i|) = 1" "("as" |x-i| =|x+i| ®x in R`
Hence, `alpha` moves on the unit circle which has center at the origin.