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

The correct option is C $z_1,z_2$ and the origin form an equilateral triangle

We have, $\frac{z_1}{z_2}+\frac{z_2}{z_1}=1$

$\Rightarrow {z_1}^2+{z_2}^2=z_1{z}_2\Rightarrow {z_1}^2+{z_2}^2+{z_3}^2=z_1{z}_2+z_2{z}_3+z_3{z}_1.$

where $z_3=0$.

$\Rightarrow z_1,z_2$ and the origin form an equilateral triangle.