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Question

If $${ z }_{ 1 },{ z }_{ 2 },{ z }_{ 3 }$$ are complex numbers such that $$\left| { z }_{ 1 } \right| =\left| { z }_{ 2 } \right| =\left| { z }_{ 3 } \right| =\left| \dfrac { 1 }{ { z }_{ 1 } } +\dfrac { 1 }{ { z }_{ 2 } } +\dfrac { 1 }{ { z }_{ 3 } }  \right| =1$$ , then $$\left| { z }_{ 1 }+{ z }_{ 2 }+{ z }_{ 3 } \right|$$ 


A
Equal to 1
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B
Greater than 3
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C
Less than 1
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D
Equal to 3
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Solution

The correct option is A Equal to 1
$$z_{1}, z_{2}, z_{3}$$ are complex numbers such that
$$|z_{1}|=|z_{2}|=|z_{3}|=\left|\dfrac{1}{z_{1}}+\dfrac{1}{z_{2}}+\dfrac{1}{z_{3}}\right|=1$$
$$\therefore z, \bar{z_{1}}=1\Rightarrow \bar{z_{1}}=\dfrac{1}{z_{1}}$$ and So on 
Also $$|z_{1}|=|\bar{z_{1}}|$$, i.e., $$\bar{z_{1}}$$ is the conjugate of $$z$$, 
$$|z_{1}+z_{2}+z_{3}|=|\overline{z_{1}+z_{2}+z_{3}}|$$
$$=|\bar{z_{1}}+\bar{z_{2}}+\bar{z_{3}}|$$
$$=\left|\dfrac{1}{z_{1}}+\dfrac{1}{z_{2}}+\dfrac{1}{z_{3}}\right|$$
$$\therefore |z_{1}+z_{2}+z_{3}|=1$$

Mathematics

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