Question

The three vertices of a triangle are represented by the complex number $0,z_1$ and $z_2$. If the triangle is equilateral, then

• A. ${z_1}^2+{z_2}^2+z_1{z}_2=0$
• B. ${z_1}^2+{z_2}^2=z_1{z}_2$
• C. ${z_1}^2-{z_2}^2=z_1{z}_2$
• D. None of these

Answer 1

Answer: (B)

${z_1}^2+{z_2}^2=z_1{z}_2$

Since the triangle is equilateral

$\therefore |z_1-0|=|z_2-z_1|=|0-z_2|$

$\Rightarrow {\left|z_1\right|}^2={|z_2-z_1|}^2={\left|z_2\right|}^2$

$\Rightarrow z_1\overline{z_1}=(z_2-z_1).(\overline{z_2}-\overline{z_1})=z_2\overline{z_2}$

Now, $z_1\overline{z_2}=z_2\overline{z_2}\Rightarrow \frac{z_1}{z_2}=\frac{\overline{z_2}}{\overline{z_1}}=\frac{z_1-z_2}{\overline{z_2}-\overline{z_1}}$

Also, $z_2\overline{z_2}=(z_2-z_1)(\overline{z_2}-\overline{z_1})=(z_2-z_1)\frac{(z_1-z_2)\overline{z_2}}{z_1}$

$\therefore z_1{z}_2=(z_2-z_1)(z_1-z_2)=z_2{z}_1-{z_1}^2-{z_2}^2+z_1{z}_2$

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