If ` z, z _1 and z_2 ` are complex numbers such that ` z = z _1 z_2 ` and `|barz_2 - z_1| le 1`, then maximum value of ` |z| -` Re(z) is _____.

Answer

0.5

Explanation

We know that
`|barz_(2) -z_(1)|^(2) =|z_(2)|^(2) +|z_(1)|^(2) - Re(z_(1)z_(2))`
`=(|z_(2)|-|z_(1)|)^(2) +2(|z|-Re(z))`
`therefore |z| -Re(z) = (|barz_(2) -barz|^(2))/(2) -((|z_(2)| -|z_(1)|)^(2))/(2) le (1)/(2) -((|z_(2)|-|z_(1)|)^(2))/(2) le (1)/(2)`
`therefore (|z|- Re(z))_("max") = (1)/(2)`