Question

Let $z_1$ be a fixed point on the circle of radius $1$ centered at the origin in the Argand plane and $z_1 ≠ ± 1$. Consider an equilateral triangle inscribed in the circle with $z_1, z_2, z_3$ as the vertices taken in the counter clockwise direction. Then $z_1z_2z_3$ is equal to

• A. $z^{2}_{1}$
• B. $z^{3}_{1}$
• C. $z^{4}_{1}$
• D. $z_{1}$

Answer 1

Answer: (B)

Given, $z_{1} \neq \pm 1$
Since, $z_{2}$ and $z_{3}$ can be obtained by rotating vector representing through $\frac{2 \pi}{3}$ and $\frac{4 \pi}{3}$, respectively
$\therefore z_{2}=z_{1} \omega $
and $z_{3}=z_{1} \omega^{2} $
$ \therefore z_{1}\, z_{2}\, z_{3} =z_{1} \times z_{1} \,\omega \times z_{1} \omega^{2}$
$=z_{1}^{3} \,\omega^{3}$
$=z_{1}^{3}$
$\left[\because \omega^{3}=1\right]$