Question

Let $z_1$ be a fixed point on the circle of radius $1$ centered at the origin in the Argand plane and $z_1=\pm 1$. Consider an equilateral triangle inscribed in the circle with $z_1,z_2,z_3$ as the vertices taken in the counter clockwise direction. The $z_1{z}_2{z}_3$ is equal to

• A. $|z_1{|}^2$
• B. $|z_1{|}^3$
• C. $|z_1{|}^4$
• D. $|z_1|$

Answer 1

Answer: (B)

$|z_1{|}^3$

Given $z^2=1$ ( circle of radius $1,$ center $(0,0)$ )

$z_1$ be point $z_1\pm 1$

Let $z_1=1$

Given $z_1,z_2,z_3$ form equilateral triangle

Inclination of $z_2,z_1$ is $150°$ with positive x axis $m=\frac{1}{\sqrt{3}}$

Equation of $z_1,z_2$ is

$\Rightarrow y-0=\frac{-1}{\sqrt{3}}(x-1)$

and ${\left|z_2\right|}^2=1$

by solving we get $(\frac{-1}{2},\frac{\sqrt{3}}{2})$

Parallely $|z_1{|}^3=(\frac{-1}{2},\frac{\sqrt{-3}}{2}).$

Hence, the answer is $(\frac{-1}{2},\frac{\sqrt{-3}}{2}).$