Question

If $z_1,z_2,z_3$ are vertices of an equilateral triangle which is inscribed in a circle $|z|=2$ and if $z_1=1+i\sqrt{3}$, then $z_2$ and $z_3$ are respectively

• A. $-2,(1-i\sqrt{3})$
• B. $2,(1+i\sqrt{3})$
• C. $-2,(1+i\sqrt{3})$
• D. $2,(1-i\sqrt{3})$

Answer 1

The correct option is A $-2,(1-i\sqrt{3})$

$z_1=1+i\sqrt{3}$
$z_1=2e^{i\left(\frac{\pi }{3}\right)}$
Since, $z_1,z_2$ and $z_3$ represent the vertices of an equilateral triangle, their arguments will differ by $\frac{2\pi }{3}$.
Hence,
$z_2=2e^{i\left(\frac{3\pi }{3}\right)}=-2$
$z_3=2e^{i\left(\frac{5\pi }{3}\right)}=1-\sqrt{3}i$