If z 1 , z 2 and z 3 are complex numbers such that | z 1 | = | z 2 | = | z 3 | = | 1 z 1 + 1 z 2 + 1 z 3 | = 1 , then | z 1 + z 2 + z 3 | is [2000]

Question

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 [2000]

Smart Achievers · Accepted Answer

(1)

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

$| z_{3} |$ $| z_{1} |$ $| \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} |$ $| z_{1} |^{2}$ $= 1 \Rightarrow z_{1} \bar{z_{1}} = 1$

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

$Now,$ $| \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} |$ $= 1 \Rightarrow$ $| \bar{z_{1}} + \bar{z_{2}} + \bar{z_{3}} |$ $= 1$

$\Rightarrow$ $| \bar{z_{1} + z_{2} + z_{3}} |$ $= 1 \Rightarrow$ $| z_{1} + z_{2} + z_{3} |$ $= 1$

Solution

Q.
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 [2000]

1 equal to 1

2 less than 1

3 greater than 3

4 equal to 3

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