Question

If $z_1,z_2,z_3,z_4,z_5$ and $z_6$ are vertices in anticlockwise direction of a regular hexagon whose circumcentre is origin and vertex $z_1=2+6i,$ then

• A. $z_4=-2+6i$
• B. $z_2=(1-3\sqrt{3})+i(3+\sqrt{3})$
• C. $z_4=-2-6i$
• D. $z_2=(1-3\sqrt{3})-i(3+\sqrt{3})$

Answer 1

Answer: (B), (C)

$z_4=-2-6i$

Given: $z_1=2+6i$


Since $|z_1|=|z_2|=|z_3|=|z_4|=|z_5|=|z_6|,$ then
$\frac{z_2}{z_1}=\left|\frac{z_2}{z_1}\right|e^{i\pi /3}$
$\Rightarrow z_2=z_1{e}^{i\pi /3}=(2+6i)(\frac{1}{2}+i\frac{\sqrt{3}}{2})$
$\Rightarrow z_2=(1+3i)(1+i\sqrt{3})=1+i\sqrt{3}+3i-3\sqrt{3}$
$\Rightarrow z_2=(1-3\sqrt{3})+i(3+\sqrt{3})$
and
$\frac{z_4}{z_1}=\left|\frac{z_4}{z_1}\right|e^{i\pi }$
$\Rightarrow z_4=z_1{e}^{i\pi }=(2+6i)(-1+i0)$
$\Rightarrow z_4=-2-6i$