Complex Numbers 2 Question 15
15. If $z _1, z _2$ and $z _3$ are complex numbers such that $\left|z _1\right|=\left|z _2\right|=\left|z _3\right|=\frac{1}{z _1}+\frac{1}{z _2}+\frac{1}{z _3}=1$, then $\left|z _1+z _2+z _3\right|$ is
(a) equal to 1
(b) less than 1
(c) greater than 3
(d) equal to 3
$(2000,2 M)$
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Answer:
Correct Answer: 15. (b)
Solution:
- Given,
$$ \left|z _1\right|=\left|z _2\right|=\left|z _3\right|=1 $$
Now,
$$ \left|z _1\right|=1 $$
$\Rightarrow$
$$ \left|z _1\right|^{2}=1 \Rightarrow z _1 \bar{z} _1=1 $$
Similarly,
$$ z _2 \bar{z} _2=1, z _3 \bar{z} _3=1 $$
Again now, $\quad \frac{1}{z _1}+\frac{1}{z _2}+\frac{1}{z _3}=1$
$\Rightarrow\left|\bar{z} _1+\bar{z} _2+\bar{z} _3\right|=1 \Rightarrow \overline{\left|z _1+z _2+z _3\right|}=1$
$\Rightarrow \quad\left|z _1+z _2+z _3\right|=1$
