Question

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

• A. $z_2 = 1 - \sqrt {3i},z_3 =-2$
• B. $z_2 = , z_3 = 1 - \sqrt {3i}$
• C. $z_2 = -1 + \sqrt {3i},z_3 =-2$
• D. none of these

Answer 1

Answer: (A)

$amp. \left(z_{1}\right)=amp. \left(1+\sqrt{3}i\right)$



$=tan^{-1}\,\sqrt{3}=\frac{\pi}{3}$
$amp. \left(z_{2}\right)=\frac{\pi}{3}-\frac{2\pi}{3}=-\frac{\pi}{3}$
and $\left|z_{2}\right|=2$
$\therefore z_{2}=2\left[cos\left(-\frac{\pi}{3}\right)+i\,sin \left(-\frac{\pi}{3}\right)\right]$
$=2\left[cos \frac{\pi}{3}-i\,sin \frac{\pi}{3}\right]$
$=2\left(\frac{1}{2}-i \frac{\sqrt{3}}{2}\right)=1-i\sqrt{3}$
$amp. \left(z_{3}\right)=\frac{\pi}{3}+\frac{2\pi}{3}=\pi$ and $\left|z_{3}\right|=2$
$\therefore z_{3}=2\left(cos\,\pi+i\,sin\,\pi\right)=2$