Question

Let $Z_1$ and $Z_2$ be two complex numbers such that $\frac{z_1}{z_2}+\frac{z_2}{z_1}=1$. Then

• 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

Answer 1

Answer: (C)

$Z_1$ , $Z_2$ and the origin form an equilateral triangle

$\frac{z_1}{z_2}+\frac{z_2}{z_1}=1$
$z_1^2-z_1{z}_2+z_2^2=1$
$z_1(z_1-z_2)+$
Now $z_1=\cdot e^{i{\theta }_1}$
$z_2=e^{i{\theta }_2}$
$\therefore e^{2i{\theta }_1}-e^{i({\theta }_1+{\theta }_2)}+e^{41{\theta }_2}=1$
$\therefore {\theta }_1-{\theta }_2=60$
$\therefore z_1,z_2$ and zero form an equilateral triangle