Question

Let $z_{1}, \, z_{2}$ and $z_{3}$ are the points on the argand plane which lie on the circle with equation $\left|z - z_{0}\right|=\frac{4}{\sqrt{3}}$ (where $z_{0}$ is the centre of the circle). If $z_{1}=0, \, z_{2}=-4$ and $z_{3}=4+3z_{0},$ then (where arg $Z \in(-\pi, \pi])$ )

• A. $\frac{\pi }{6}$
• B. $\frac{5 \pi }{12}$
• C. $\frac{5 \pi }{6}$
• D. $\frac{2 \pi }{3}$

Answer 1

Answer: (C)

Here $\frac{z_{1} + z_{2} + z_{3}}{3}=z_{0}\Rightarrow z_{3}=4+3z_{0}$
Therefore, center coincides with the circumcentre
$\Rightarrow $ Triangle is equilateral $\Rightarrow \left|z_{1} - z_{2}\right|=4$
Clearly, $z_{3}$ either lie in the second or third quadrant
So the centre $z_{0}$ also lies in the second or third quadrant.
$\therefore z_{0}$ can be $=-2+\frac{2}{\sqrt{3}}\text{i},-2-\frac{2}{\sqrt{3}}\text{i}$
$\Rightarrow arg\left(z_{0}\right)=\frac{5 \pi }{6},\frac{7 \pi }{6}$