Question

State whether the following statement is true or false.

If the complex numbers $Z_1,Z_2$ and origin form an isosceles triangle with vertical angle $\left(2\pi /3\right)$, then $(Z_1{)}^2+(Z_2{)}^2+Z_1{Z}_2=0$

• A. True
• B. False

Answer 1

The correct option is A True

From Rotation Theorem(Coni Method), we have,

$\frac{z_1-0}{z_2-0}=\frac{OA}{OB}{e}^{2\pi i/3}$

$⟹\frac{z_1}{z_2}=\cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3}$

$⟹\frac{z_1}{z_2}=-\frac{1}{2}+i\frac{\sqrt{3}}{2}$

$⟹\frac{z_1}{z_2}+\frac{1}{2}=\frac{i\sqrt{3}}{2}$

On squaring both sides, we get,

$\frac{z_1^2}{z_2^2}+\frac{1}{4}+\frac{z_1}{z_2}=-\frac{3}{4}$

$⟹\frac{z_1^2}{z_2^2}+\frac{z_1}{z_2}+1=0$

$\therefore z_1^2+z_2^2+z_1{z}_2=0.$