Question

If `z_1, z_2 , z_3` are complex numbers such that `|z_1| = |z_2| = |z_3| = 1 |1/z_1 + 1/ z_2 + 1/ z_3| = 1`, Then find the value of `|z_1 + z_2 + z_3|` is :

Answer 1

Here, `|z_1| = |z_2| = |z_3| = 1`
As `|z_1| = 1=>z_1.bar(z_1) = 1`
`bar(z_1) = 1/z_1`
Similarly, `bar(z_2) = 1/z_2`
`bar(z_3) = 1/z_3`.
We are given,
`|1/z_1+1/z_2+1/z_3| = 1`
`:. |bar(z_1)+bar(z_2)+bar(z_3)| = 1`
Also, we know,
`|z| = |barz|`
`:.|bar(bar(z_1)+bar(z_2)+bar(z_3))| = 1`
`:. |z_1+z_2+z_3| = 1`