Question

If $z_1$ and $z_2$ be the $n$th roots of unity which subtend 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

The correct option is D

$4k$

Explanation for the correct option.

Find the representation of $n$.

It is given that $z_1$ and $z_2$ are the $n$th roots of unity. So, ${z_1}^n={z_2}^n=1$.

Also, $z_1$ and $z_2$ subtend right angle at origin. So, $\arg \left(\frac{z_1}{z_2}\right)=\frac{\mathrm{\pi }}{2}$. Thus,

$\begin{array}{rcl}\frac{z_1}{z_2}& =& \cos \frac{\mathrm{\pi }}{2}+i\sin \frac{\mathrm{\pi }}{2}\\ & =& 0+i·1\left[\cos \frac{\mathrm{\pi }}{2}=0,\sin \frac{\mathrm{\pi }}{2}=1\right]\\ & =& i\end{array}$.

Thus, ${\left(\frac{z_1}{z_2}\right)}^n=i^n$ but ${\left(\frac{z_1}{z_2}\right)}^n=1$ because ${z_1}^n={z_2}^n=1$.

Thus, $i^n=1$. This is true only when $n$ is of the form $4k$.

So, $n$ must be of the form $4k$.

Hence, the correct option is D.