If z1,z2 and z3 are complex numbers such that z1=z2=z3=1z1+1z2+1z3=1, then z1+z2+z3 is

z1=z2=z3=1now, z1=1⇒z12=1⇒z1z¯1=1⇒z¯1=1z1Similarly, z2z¯2=1 and z3z¯3=1again ,1z1+1z2+1z3=1⇒z¯1+z¯2+z¯3=1 ⇒z1+z2+z3=1 ⇒z1+z2+z3=1

If z1,z2 and z3 are complex numbers such that z1=z2=z3=1z1+1z2+1z3=1, then z1+z2+z3 is

z1=z2=z3=1now, z1=1⇒z12=1⇒z1z¯1=1⇒z¯1=1z1Similarly, z2z¯2=1 and z3z¯3=1again ,1z1+1z2+1z3=1⇒z¯1+z¯2+z¯3=1 ⇒z1+z2+z3=1 ⇒z1+z2+z3=1

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If $z_{1}, z_{2} and z_{3}$ are complex numbers such that $|z_{1}| = |z_{2}| = |z_{3}| = |\frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}}| = 1,$ then $|z_{1} + z_{2} + z_{3}|$ is

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equal to 1

less than 1

greater than 3

equal to 3

answer is A.

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Detailed Solution

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

$\begin{array}{l} now, |z_{1}| = 1 \Rightarrow {|z_{1}|}^{2} = 1 \\ \Rightarrow z_{1} {\bar{z}}_{1} = 1 \Rightarrow {\bar{z}}_{1} = \frac{1}{z_{1}} \end{array}$

Similarly, $z_{2} {\bar{z}}_{2} = 1 and z_{3} {\bar{z}}_{3} = 1$

$\begin{array}{l} again, |\frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}}| = 1 \\ \Rightarrow |{\bar{z}}_{1} + {\bar{z}}_{2} + {\bar{z}}_{3}| = 1 \\ \Rightarrow |z_{1} + z_{2} + z_{3}| = 1 \\ \Rightarrow |z_{1} + z_{2} + z_{3}| = 1 \end{array}$