Question

lf one of the vertices of the square circumscribing the circle $|z-1|=\sqrt{2}$ is $2+\sqrt{3}i$. Find the other vertices of square.

• A. $(1-\sqrt{3})+2i$ and $(1+\sqrt{3})-2i$
  
• B. $(1-\sqrt{3})$ and $(1+\sqrt{3})$
• C. $(1-\sqrt{3})+i$ and $(1+\sqrt{3})-i$
• D. None of the above

Answer 1

Answer: (C)

$(1-\sqrt{3})+i$ and $(1+\sqrt{3})-i$

Solution:

Since the centre of the circle i.e.$(1,0)$ is also the mid-point of the diagonals of square.

$⟹\frac{z_1+z_2}{2}=z_0$

or, $z_2=2-(2+i\sqrt{3})=-\sqrt{3}i$

and $\frac{z_3-1}{z_1-1}=e^{\pi /2}$

or, $z_3-1=i(z_1-1)=-\sqrt{3}+i$

or, $z_3=(1-\sqrt{3})+i$

and $\frac{z_4-1}{z_1-1}=e^{-\pi /2}$

or, $z_4-1=-i(z_1-1)=\sqrt{3}-i$

or, $z_4=(1+\sqrt{3})-i$

$⟹$ other vertices are:

$\therefore z_3=(1-\sqrt{3})+i,z_4=(1+\sqrt{3})-i$

Hence, C is the correct option.