Question

If $z_1$ 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 the points represented by $z_1$ and $z_2$

• A. lie on a straight line
• B. form a right angled triangle
• C. form an equilateral triangle
• D. from an isosceles triangle

Answer 1

Answer: (C)

form an equilateral triangle

Answer 2

We know that, if $z_{1}, z_{2}$ and $z_{3}$ are the vertices of an equilateral triangle. Then,
$z_{1}^{2}+z_{2}^{2}+z_{3}^{2}-z_{1} z_{2}-z_{2} z_{3}-z_{3} z_{1}=0\ldots$(i)
Now, but we have
$\frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{1}} =1$
$\Rightarrow z_{1}^{2}+z_{2}^{2} =z_{1} z_{2}$
$\Rightarrow z_{1}^{2}+z_{2}^{2}-z_{1} z_{2} =0$
Here, $z_{3}=0$
Hence, given points form an equilateral triangle.