It’s complex feeling!

Example.

If z_{1}, z_{2}, 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

(a)  equal to 1

(b) less than 1

(c) greater than 3

(d) equal to 3

Solution:

Since |z_{1}|=|z_{2}|=|z_{3}|=1, we get z_{1} \overline{z_{1}}=z_{2} \overline{z_{2}}=z_{3} \overline{z_{3}}=1

Now, 1=|\frac{1}{z_{1}}+\frac{1}{z_{2}}+\frac{1}{z_{3}}|=|\overline{z_{1}}+\overline{z_{2}}+\overline{z_{3}}|=|\overline{z_{1}+z_{2}+z_{3}}|

\Longrightarrow 1 = |z_{1}+z_{2}+z_{3}|

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