Question

Let $z_1$ and $z_2$ be $n$th roots of unity which subtend a right angle at the origin. Then $n$ must be of the form

• A. $4k+1$
• B. $4k$
• C. $4k+3$
• D. $4k+2$

Answer 1

Answer: (B)

$4k$

Solution:

Given that:

$arg\left(\frac{z_1}{z_2}\right)=\frac{\pi }{2}$

We know that, $arg\left(i\right)=\frac{\pi }{2}$

$arg\left(\frac{z_1}{z_2}\right)=\frac{\pi }{2}=arg\left(i\right)$

$⟹\frac{z_1}{z_2}=i$

$⟹{\left(\frac{z_1}{z_2}\right)}^n=i^n$

$⟹1=i^n$ ... $[\because z_1,z_2$ are $n$th root of unity $]$

Now, $i=\sqrt{-1}⟹i^4=i^8=i^{12}=....=1$

$⟹i^{4k}=i^n$

$⟹n=4k$

Hence, B is the correct option.