Let $Z_{1}, Z_{2}, Z_{3}$ be three non zero complex numbers such that a= $|Z_{1}|$ , b= $|Z_{2}|$ , c= $|Z_{3}|$ if the determinant
$|\begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix}| = 0$

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$|Z_{1}| + |Z_{2}| + |Z_{3}| = 0$

$|Z_{1}| + |Z_{2}| + |Z_{3}| = a b c$

$|Z_{1}| = |Z_{2}| = |Z_{3}| = a b c$

$|z_{1 -} z_{2}| = |z_{2 -} z_{3}|$

answer is B.

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Detailed Solution

$|\begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix}| = 0 C_{1} \Rightarrow C_{1} + C_{2} + C_{3} \Rightarrow (a + b + c) |\begin{matrix} 1 & b & c \\ 1 & c & a \\ 1 & a & b \end{matrix}| = 0 \begin{aligned} (a + b + c) (a^{2} + b^{2} + c^{2} - a b - b c - c a) = 0 \\ \frac{1}{2} (a + b + c) [(a - b)^{2} + (b - c)^{2} + (c - a)^{2}] = 0 \\ a + b + c = 0 o r (a - b)^{2} + (b - c)^{2} + (c - a)^{2} = 0 \end{aligned} \Rightarrow a + b + c = 0 o r a = b = c ∴ |z_{1}| + |z_{2}| + |z_{3}| = 0 o r |Z_{1}| = |Z_{2}| = |Z_{3}|$