Question

Let `z` be not a real number such that `(1+z+z^2)//(1-z+z^2) in R ,` then prove tha `|z|=1.`

Answer 1

` (1 + z + z ^(2))/(1 - z + z ^(2) ) = (1 - z + z ^(2) + 2 z )/( 1 - z + z ^(2) ) ` ltbrrgt ` " " = 1 + (2)/( z - 1 + (1//z)) ` is real
Hence, ` z + (1)/(z) in R `
` rArr z + (1)/(z) = bar z + (1)/(z)`
` rArr (z - bar z) = (1)/(barz) - (1)/(z) = (z - bar z)/(|z|^(2)) `
` rArr (z - bar z ) (|z|^(2) - 1 ) = 0 `
` rArr |z| = 1 ( z = bar z ` is not possible as z is not real )