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. form an isosceles triangle

Answer 1

The correct option is C form an equilateral triangle

$z_1=z_2{e}^{i\pi /3}$

$2z_1=z_2(1+i\sqrt{3})$

$2z_1-z_2=i\sqrt{3}{z}_2$

$\Rightarrow 4z_1^2+z_2^2-4z_1{z}_2=-3z_2^2$

$\Rightarrow 4z_1^2+4z_2^2=4z_1{z}_2$

Hence form equilateral triangle