Question

Let $z_1$,$z_2$,$z_3$are vertices of an equilateral triangle circumscribing the circle $\left|z\right|=1$,if $z_1=1+i\sqrt{3}$ and$z_1$,$z_2$,$z_3$ are in anticlockwise sense, then $z_2$ equals

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

Answer 1

Answer: (B)

$-2$

This is a circle with centre at origin and radius $=1$

$|z|=1$ .

$z_1=1+i\sqrt{3}=2(cos\frac{\pi }{3}+isin\frac{\pi }{3})=2e^{i\pi /3}$

From figure, $z_2$ can be obtained by rotating $z_1$ by ${120}^{\circ }$

$\therefore z_2=z_1{e}^{i2\pi /3}=2e^{\pi }=2(cos\pi +isin\pi )=-2$