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 + 2
• C. 4k + 3
• D. 4k

Answer 1

The correct option is D 4k

$Z_1=e^{i\frac{2k_1\pi }{n}}$
$Z_2=e^i\frac{2k_2\pi }{n}$
Now, $Z_1,Z_2$ subtend a right angle at origin
$\Rightarrow \frac{Z_1}{|Z_1|}=\frac{Z_2}{|Z_2|}{e}^{i\left(\frac{\pi }{2}\right)}$

$\Rightarrow e^{i(k_1-k_2)\frac{2\pi }{n}}=e^{i\left(\frac{\pi }{2}\right)}$

Hence,

$(k_1-k_2)\frac{2\pi }{n}=\frac{\pi }{2}$

$\Rightarrow k_1-k_2=\frac{n}{4}$

As $k_1,k_2$ are integers, $n$ must be of the form $4k$