Question

a, b, c are three complex numbers on the unit circle |z| = 1, such that abc = a + b + c. Then |ab + bc + ca| is equal to-

• A. 3
• B. 6
• C. 1
• D. 2

Answer 1

Answer: (C)

1

Answer 2

Sol. a$\bar{a}$= b$\bar{b}$= c$\bar{c}$= 1 \ $\bar{a}$= $\frac{1}{a}$ etc.

| abc | = |a + b + c| = |$\bar{a}$+$\bar{b}$+$\bar{c}$ |

= $\left| \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right|$=$\left| \frac{\sum_{}^{}{ab}}{abc} \right|$

\ |$\sum_{}^{}{ab}$| = |abc| |abc| = (|a| |b| |c|)² = 1