Question

Let $z_1$ and $z_2$ be two non-zero complex numbers such that $\frac{z_1}{z_2}+\frac{z_2}{z_1}=1$, then the origin and points represented by $z_1$ and $z_2$

• A. lie on a straight line
• B. form a right triangle
• C. form an equilateral triangle
• D. none of these

Answer 1

Answer: (C)

Let $z=\frac{z_1}{z_2}$, then $z+\frac{1}{z}=1\Rightarrow z^2-z+1=0$ $\Rightarrow z=\frac{1\pm \sqrt{3}i}{2}$
$\Rightarrow \frac{z_1}{z_2}=\frac{1\pm \sqrt{3}i}{2}$
If $z_1$ and $z_2$ are represented by $A$ and $B$ respectively and $O$ be the origin, then
$\frac{OA}{OB}=\frac{|z_1|}{|z_2|}=\left|\frac{1\pm \sqrt{3}i}{2}\right|=\sqrt{\frac{1}{4}+\frac{3}{4}}=1$
$\Rightarrow OA=OB$
$\text{ Also, }\frac{AB}{OB}=\frac{|z_2-z_1|}{|z_2|}=\left|1-\frac{z_1}{z_2}\right|$
$=\left|1-\left(\frac{1}{2}\pm \frac{\sqrt{3}}{2}i\right)\right|=\left|\frac{1}{2}\mp \frac{\sqrt{3}}{2}i\right|=\sqrt{\frac{1}{4}+\frac{3}{4}}=1$
$\Rightarrow AB=OB$
$\text{ Thus, }OA=OB=AB.$
$\therefore △OAB$ is an equilateral triangle.