Question

z_{​​​​​​1} ,z_{​​​​​​2} , z_{​​​​​​3} are complex numbers such that the modulus of each number is equal to the modulus of the sum of reciprocals of these numbers , which is equal to 1. Find the modulus of sum of these three numbers

Answer 1

Question kind of looks complex but it's very simple here's answer:

| Z1 | = | Z2 | = | Z3 | = 1 = | 1/Z1 + 1/Z2 + 1/Z3 |

So value for Z1,Z2,Z3 must be 1 or -1 and from| 1/Z1 + 1/Z2 + 1/Z3 | we can assume that only one value from these three constant is -1 .

1. | 1/1 + 1/-1 + 1/1 | = | 1 - 1 + 1 |
2. = 1.
3. So one value must be negetive.

You can take any once value as -1 and other's as 1 then,

Suppose take z2= -1 then z1 = z3 = 1 as above said so,

| Z1 + Z2 + Z3 | = | 1 - 1 + 1 | = | 2 - 1 | = 1.

ALSO OTHER POSSIBILITY IS TWO OF VALUE CAN BE -1 AND ONE VALUE 1 THEN ALSO YOU GET SAME ANSWER.