Question

Let $A\left(z_1\right)$ and $B\left(z_2\right)$ are points on Argand plane. If $\frac{z_1}{z_2}+\frac{z_2}{z_1}=1$, then

• A. origin, $A$ and $B$ are collinear.
• B. origin, $A$ and $B$ are vertices of an equilateral triangle.
• C. origin, $A$ and $B$ are vertices of a right angled triangle.
• D. origin, $A$ and $B$ are vertices of an isosceles triangle.

Answer 1

The correct option is B origin, $A$ and $B$ are vertices of an equilateral triangle.

Given condition:
$\frac{z_1}{z_2}+\frac{z_2}{z_1}=1\Rightarrow z_1^2+z_2^2=z_1{z}_2\Rightarrow z_1^2+z_2^2+0^2=z_1{z}_2+z_2\cdot 0+0\cdot z_1\cdots \left(1\right)$

We know that $z_1,z_2$ and $z_3$ are vertices of a an equilateral triangle, if
$z_1^2+z_2^2+z_3^2=z_1{z}_2+z_2{z}_3+z_3{z}_1$

Therefore, from equation $\left(1\right)$, origin, $A\left(z_1\right)$ and $B\left(z_2\right)$ are vertices of an equilateral triangle.