Question

Let $z_{1},z_{2},z_{3}$ be three vertices of an equilateral triangle circumscribing the circle $|z| = \frac{1}{2}.$ If $z_{1} = \frac{1}{2} + \frac{\sqrt{3}i}{2}$ and $z_{1},z_{2},z_{3}$ are in anticlockwise sense then $z_{2}$ is

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

Answer 1

Answer: (D)

– 1

Answer 2

Sol. $z_{2} = z_{1}e^{i2\pi/3} = \left( \frac{1}{2} + \frac{\sqrt{3}}{2}i \right)$ $\left( \cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3} \right)$

$\mathbf{=}\left( \frac{\mathbf{1}}{\mathbf{2}}\mathbf{+}\frac{\sqrt{\mathbf{3}}}{\mathbf{2}}\mathbf{i} \right)\left( \frac{\mathbf{- 1}}{\mathbf{2}}\mathbf{+}\frac{\sqrt{\mathbf{3}}}{\mathbf{2}}\mathbf{i} \right)$ $\mathbf{=}\frac{\mathbf{-}\mathbf{3}}{\mathbf{4}}\mathbf{-}\frac{\mathbf{1}}{\mathbf{4}}\mathbf{=}\mathbf{-}\mathbf{1.}$