    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

$4k+1$

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B

$4k+2$

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C

$4k+3$

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D

$4k$

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Solution

## 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, $\mathrm{arg}\left(\frac{{z}_{1}}{{z}_{2}}\right)=\frac{\mathrm{\pi }}{2}$. Thus,$\begin{array}{rcl}\frac{{z}_{1}}{{z}_{2}}& =& \mathrm{cos}\frac{\mathrm{\pi }}{2}+i\mathrm{sin}\frac{\mathrm{\pi }}{2}\\ & =& 0+i·1\left[\mathrm{cos}\frac{\mathrm{\pi }}{2}=0,\mathrm{sin}\frac{\mathrm{\pi }}{2}=1\right]\\ & =& i\end{array}$.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.  Suggest Corrections  11      Similar questions
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