Question

Let $ {z_1} $ and $ {z_2} $ be the nth roots of unity which subtend a right angle at the origin. Then n must be of the form
A.4k+1
B.4k+2
C.4k+3
D.4k

Answer 1

Hint:In order to solve this question, we will use the concept of nth roots of unity and we will use deMoivre’s theorem. We will start by writing the given equation in the polar form, then we will substitute the angle as given in the question and then solve step by step to reach the answer.

Complete step-by-step answer:
In the nth root of unity, the positive value of integer n, are the n roots of the unity.
 $ {Z^n} = 1 $
In polar form the above equation can be written as
 $
  {Z^n} = \cos \left( {0 + 2k\pi } \right) + i\sin \left( {0 + 2k\pi } \right) \\
  {Z^n} = {e^{i2k\pi }} \\
  where\,\,k = 0,1,2,3.......... \\
 $
Using the above equation, we will find the roots of unity
As we know
 $
   \Rightarrow {Z^n} = 1 \\
   \Rightarrow {Z^n} - {1^n} = 0 \\
  $
Since $ {z_1} $ and $ {z_2} $ are the roots of unity; substitute them in above equation
 $ $ $
   \Rightarrow Z_1^n = 1 \\
   \Rightarrow Z_1^n - {1^n} = 0 \\
 $
Using DeMoivre’s theorem
 $ \Rightarrow {Z_1} = {e^{\dfrac{{i2\pi }}{n}}} $
Similarly,
 $
   \Rightarrow Z_2^n = 1 \\
   \Rightarrow Z_2^n - {1^n} = 0 \\
 $
 $ \Rightarrow {Z_2} = {e^{\dfrac{{i2\pi }}{n}}} $
The argument of the above complex numbers is
 $ {\theta _1} = \dfrac{{2\pi k}}{n} $
Therefore the argument of the $ {z_2} $ is
 $ {\theta _2} = \dfrac{{2\pi \left( {k + 1} \right)}}{n} $
It is given that the nth roots of unity subtends 90 degree at the origin, therefore the difference between the arguments of the two roots must be equal to 90 degree
 $ \Rightarrow {\theta _2} - {\theta _1} = \dfrac{\pi }{2} $
Substituting the value of $ {\theta _1} = \dfrac{{2\pi k}}{n} $ and $ {\theta _2} = \dfrac{{2\pi \left( {k + 1} \right)}}{n} $ in the above equation, we get
 $
   \Rightarrow {\theta _2} - {\theta _1} = \dfrac{\pi }{2} \\
   \Rightarrow \dfrac{{2\pi \left( {k + 1} \right)}}{n} - \dfrac{{2\pi k}}{n} = \dfrac{\pi }{2} \\
   \Rightarrow 2\dfrac{\pi }{n}\left( {k + 1 - k} \right) = \dfrac{\pi }{2} \\
   \Rightarrow n = 4 \\
 $

Therefore the order of the given number is of 4k
Hence, the correct option is D.

Note:As we know, the complex numbers are the points in the complex plane and nth roots have the magnitude of 1, all the nth roots lie inside the circle of radius 1 in a complex plane. In order to solve these types of problems, you need to remember the De Moivre’s theorem. The roots of unity are used in number theory, group theory and discrete fourier transform etc.