Question

If one of the vertices of the square, inscribed in the circle $|z-1|=2$, is $2+\sqrt{3}i$, then other vertices of the square are

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

Answer 1

Answer: (B), (C), (D)

The correct options are
B $(1-\sqrt{3})+i$
C $(\sqrt{3}+1)-i$
D $-i\sqrt{3}$

The given circle is $|z-1|=2$ where $z_0=1$ is the centre and $2$ is radius of the circle. $z_1$ is one of the vertices of the square inscribed in the given circle.

Hence all the other vertices can be obtained by rotating the $z_1$ about $z_0$ by $\pm \frac{\pi }{2},\pi$. Thus,
$z_2-z_0=(z_1-z_0)e^{i\pi /2}$
$\Rightarrow z_2-1=(2+i\sqrt{3}-1)\left(i\right)$
$\Rightarrow z_2=i-\sqrt{3}+1$
$\Rightarrow z_2=(1-\sqrt{3})+i$
similarly $z_4=(\sqrt{3}+1)-i$
and $z_3-z_0=(z_1-z_0)e^{i\pi }$
$\Rightarrow z_3=z_0-(z_1-z_0)=-i\sqrt{3}$