A) \[{{z}_{1}},{{z}_{2}}\]are collinear
B) \[{{z}_{1}},{{z}_{2}}\]and the origin form a right angled triangle
C) \[{{z}_{1}},{{z}_{2}}\]and the origin form an equilateral triangle
D) None of these
Correct Answer: C
Solution :
We have \[\frac{{{z}_{1}}}{{{z}_{2}}}+\frac{{{z}_{2}}}{{{z}_{1}}}=1\Rightarrow z_{1}^{2}+z_{2}^{2}={{z}_{1}}{{z}_{2}}\] Þ \[z_{1}^{2}+z_{2}^{2}+z_{3}^{2}={{z}_{1}}{{z}_{2}}+{{z}_{1}}{{z}_{3}}+{{z}_{2}}{{z}_{3}},\]where\[{{z}_{3}}=0\] Þ \[{{z}_{1}},{{z}_{2}}\] and the origin \[(\because {{z}_{3}}=0)\] form an equilateral triangle.
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