Question

$z_1,z_2$ are two non-real complex numbers such that $\frac{z_1}{z_2}+\frac{z_2}{z_1}=1$. Then $z_1,z_2$ and the origin

• A. are collinear
• B. form right angled triangle
• C. form right angle isosceles triangle
• D. form an equilateral triangle

Answer 1

The correct option is D form an equilateral triangle

If $\frac{z_1}{z_2}=z$, the given equation becomes

$z^2-z+1=0\Rightarrow z=-\omega$ and $-{\omega }^2$

$\Rightarrow \frac{z_1}{z_2}=-\omega \Rightarrow z_1=-z_2\omega$

$OB=|z_2-0|=\left|z_2\right|$

$OA=|z_1-0|=|-z_2\omega |$

$=\left|z_2\right||-\omega |=\left|z_2\right|$

and $AB=|z_2-z_1|=|z_2+z_2\omega |$

$=\left|z_2\right||1+\omega |=\left|z^2\right||-{\omega }^2|=\left|z_2\right|$

Thus $z_1,z_2$ and origin form an equilateral triangle.