More complex stuff

Problem.

If z_{1}, z_{2}, \ldots , z_{n} lie on the unit circle |z|=2, then value of

E=|z_{1}+z_{2}+\ldots+z_{n}|-4|\frac{1}{z_{1}}+\frac{1}{z_{2}}+\ldots+\frac{1}{z_{n}}| is

(a) 0

(b) n

(c) -n

(d) none of these.

Solution.

As z_{1},z_{2},\ldots, z_{n} lie on the circle |z|=2, |z_{i}|=2 \Longrightarrow |z_{i}|^{2}=4 \Longrightarrow z_{i}\overline{z_{i}}=4 for i=1,2,3, \ldots, n

Thus, \frac{1}{z_{i}}=\frac{\overline{z_{i}}}{4} for i=1, 2, 3, \ldots, n

Hence, E=|z_{1}+z_{2}+\ldots+z_{n}|-4|\frac{\overline{z_{1}}}{4}+\frac{\overline{z_{2}}}{4}+\ldots+\frac{\overline{z_{n}}}{4}|, which in turn equals

|z_{1}+z_{2}+\ldots+z_{n}|-|\overline{z_{1}}+\overline{z_{2}}+\ldots+\overline{z_{3}}|, that is,

|z_{1}+z_{2}+\ldots+z_{n}|-|\overline{z_{1}+z_{2}+\ldots+z_{n}}|=0.

(since |z|=|\overline{z}|).

Answer. Option a.

More later,

Nalin Pithwa

 

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