$|z-1|=\sqrt{2}$ is a circle inscribed in a square whose one vertex is $2+i \sqrt{3}$. Which of the

Question

$|z-1|=\sqrt{2}$ is a circle inscribed in a square whose one vertex is $2+i \sqrt{3}$. Which of the following is not among the remaining vertices?

Professor PC Thomas Classes · Answer

Circle is $|z-1|=\sqrt{2}$ By rotation $\frac{z_{2}-z_{0}}{z_{1}-z_{0}}=\frac{O B}{O A} e^{ \pm \frac{\pi i}{2}}= \pm i$\ $	herefore \quad z_{2}=z_{0} \pm i\left(z_{1}-z_{0}ight)$\ $=1 \pm i(1+i \sqrt{3})=1 \pm(i-\sqrt{3})$\ $=(1-\sqrt{3})+i,(1+\sqrt{3})-i$\ $\frac{z_{1}+z_{3}}{2}=z_{0}=1 . \quad 	herefore \quad z_{3}=-\sqrt{3} i$\ Thus the remaining vertices are\ $(1+\sqrt{3})-i,-\sqrt{3} i,(1-\sqrt{3})+i$\