Question

If |z₁| = |z₂| = |z₃| 1 and z₁, z₂, z₃ represents the vertices of an equilateral triangle, then (a) z₁ + z₂ + z₃ = 0 and z₁ z₂ z₃ =1 (b) z₁ + z₂ + z₃ = 1 and z₁ z₂ z₃ =1
(c) z₁z₂ + z₂z₃ + z₃z₁ = 0 and z₁ + z₂ + z₃ = 0 (d) z₁z₂ + z₂z₃ + z₃z₁ = 0 andz₁ z₂ z₃ =1​

Answer 1

Since ∆ ABC is equilateral, therefore, BC when rotated through 60° coincides with BA as per fig. But to turn the direction of a complex number through an ∠⊖, we multiply it by cos ⊖ + i sin ⊖.

⸫ BC⃗ (cosπ / 3 + i sinπ / 3) = BA⃗

(z₃ - z₂)(1 + i √3 /2) = z₁ - z₂

OR

i √3 (z₃ - z₂) = 2z₁ - z₂ - z₃

Squiaring,

- 3 (z₃ - z₂)² = (2z₁ - z₂ - z₃)²

OR

4(z₁² + z₂² + z₃² - z₁z₂ - z₂z₃ - z₃z₁) = 0

whence follows the required condition.

Answer 2

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