Question

If $|z_1|=|z_2|=|z_3|=1$ and $z_1,z_2,z_3$ are represented by the vertices of an equilateral triangle. Then

• A. $z_1+z_2+z_3$
• B. $z_1{z}_2+z_2{z}_3+z_3{z}_1=1$
• C. $z_1+z_2+z_3=1$
• D. none of these

Answer 1

The correct option is D none of these

Angle subtended by a chord on centre is double of that subtended on circumference by the same chord.
Solution: $z_2=z_1{e}^{2\pi i/3}=z_{1}w$




$z_3=z_1{e}^{4\pi i/3}=z_1{w}^2$
$z_1{z}_2{z}_3=z_1^{3}1.w.w^2=z_1^3$
$z_1+z_2+z_3=z_1(1+w+w^2)=0$
$z_1{z}_2+z_2{z}_3+z_3{z}_1=z_1^2(w+w^3+w^2)=0$