Complex Numbers 2 Question 35
36. If the complex numbers, $z _1, z _2$ and $z _3$ represent the vertices of an equilateral triangle such that $\left|z _1\right|=\left|z _2\right|=\left|z _3\right|$, then $z _1+z _2+z _3=0$.
(1984, 1M)
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Solution:
- Since, $z _1, z _2, z _3$ are vertices of equilateral triangle and $\left|z _1\right|=\left|z _2\right|=\left|z _3\right|$
$\Rightarrow z _1, z _2, z _3$ lie on a circle with centre at origin.
$\Rightarrow$ Circumcentre $=$ Centroid
$$ \begin{array}{lrl} \Rightarrow & 0 & =\frac{z _1+z _2+z _3}{3} \\ \therefore & z _1+z _2+z _3 & =0 \end{array} $$
