Question

If complex numbers $ {{z}_{1}},{{z}_{2}} $ and $ {{z}_{3}} $ represents the vertices of an equilateral triangle such that $ |{{z}_{1}}|=|{{z}_{2}}|=|{{z}_{3}}|, $ then

• A. $ {{z}_{1}}+{{z}_{2}}+{{z}_{3}}\le 0 $
• B. $ {{z}_{1}}+{{z}_{2}}+{{z}_{3}}\ge 0 $
• C. $ {{z}_{1}}+{{z}_{2}}+{{z}_{3}}=0 $
• D. None of these

Answer 1

Answer: (C)

Given, $ |{{z}_{1}}|=|{{z}_{2}}|=|{{z}_{3}}| $ Three points lies on the circle, whose centre is origin. Since, these points are vertices of an equilateral triangle so orthocentre and circumcentre will be same.
Hence, $ \frac{{{z}_{1}}+{{z}_{2}}+{{z}_{3}}}{3}=0 $
$ \Rightarrow $ $ {{z}_{1}}+{{z}_{2}}+{{z}_{3}}=0 $