Show that: , where are the roots of unity and are any t…

Maths Complex Number The Modulus and the Conjugate of a Complex Number Subjective Type

Published on: August 12, 2026

Show that: , where are the roots of unity and are any two complex numbers.

Text Solution

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The correct answer is:

CHECK THE SOLUTION.

Sol. We know | α r | = 1; r = 0, 1, 2, ……, (n –1)

and 1 + α + α 2 + ……..+ α n–1 = 0

i.e.

Further for any two complex numbers A and B,

|A + B| 2 = |A| 2 + |B| 2 + B + A

So, |z 1 + α r z 2 | 2 = |z 1 | 2 + | α r z 2 | 2 + α r z 2 + z 1

= |z 1 | 2 + |z 2 | 2 + α r , z 2 + z 1 α –r

∴ = n |z 1 | 2 + n|z 2 | 2

+ z 2

= n

Since,

⇒

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