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
Answers
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
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
Explanation:
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Clearly z
1
,z
2
,z
3
lie on unit circle ∣z∣=1
It is known that angle subtended at circumference must be half of that subtended at centre.
Hence the 3 points are separated by 120
∘
So, z
1
+z
2
+z
3
=e
iθ
+e
i(θ+
3
2π
)
+e
i(θ+
3
4π
)
We know e
iθ
=cosθ+isinθ
Using this expression z
1
+z
2
+z
3
gives value 0.
