Complex Numbers 4 Question 17
17. Prove that the complex numbers $z _1, z _2$ and the origin form an equilateral triangle only if $z _1^{2}+z _2^{2}-z _1 z _2=0$.
$(1983,2 M)$
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Solution:
- Since, $z _1, z _2$ and origin form an equilateral triangle.
$\because$ if $z _1, z _2, z _3$ from an equilateral triangle, then
$$ \begin{array}{rlrl} & & z _1^{2}+z _2^{2}+z _3^{2} & =z _1 z _2+z _2 z _3+z _3 z _1 \\ & \Rightarrow & z _1^{2}+z _2^{2}+0^{2} & =z _1 z _2+z _2 \cdot 0+0 \cdot z _1 \\ \Rightarrow & & z _1^{2}+z _2^{2} & =z _1 z _2 \\ \Rightarrow & z _1^{2}+z _2^{2}-z _1 z _2 & =0 \end{array} $$
