Question

Let $z_1,z_2,z_3$ are the vertices of an equilateral triangle inscribed in the circle $\left|z\right|$ = 2. If $z_1,z_2,z_3$ are in the clockwise sense and $z_1$ = $1+i\sqrt{3}$, then:

• A. =
• B. = , = 2
• C. = , = -2
• D. = , = 2

Answer 1

The correct option is A

=

$\left|z_1\right|=\left|z_2\right|=\left|z_3\right|=2$

$\frac{Z_1}{Z_2}$ = $\left|\frac{Z_1}{Z_2}\right|e^{2\pi i/3}$

$\Rightarrow Z_2$ = $(1+i\sqrt{3})e^{-2\pi i/3}$

$\Rightarrow Z_2$ = $(1+i\sqrt{3})⟮-\frac{1}{2}-\frac{i\sqrt{3}}{2}⟯$

$\Rightarrow Z_2$ = $1-\sqrt{3}i$

$\Rightarrow \frac{Z_3}{Z_2}$ = $\left|\frac{Z_1}{Z_2}\right|e^{2\pi i/3}$

$\Rightarrow Z_3$ = $(1-\sqrt{3}i)⟮-\frac{1}{2}+i\frac{\sqrt{3}}{2}⟯$ = -2