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Question

Let z1,z2,z3​ be three distinct complex numbers lying on a circle whose centre is at the origin. If zi+zjzk, ​where i,j,k{1,2,3} and ijk are real numbers, then the value of 4(z1×z2×z3) is

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Solution

As z1,z2,z3 ​lie on the circle centered at origin.
|z1|=|z2|=|z3|=r
(where r is the radius of the circle)
Let z1=reiθ1
z2=reiθ2
z3=reiθ3
​It is given that z1+z2z3R
z1=¯¯¯¯¯¯¯¯¯z2z3reiθ1=reiθ2reiθ3r=r2ei(θ1+θ2+θ3)rei(θ1+θ2+θ3)=1 (r0)
Taking mod both sides, we get
r=1

Hence, 4(z1×z2×z3)
=4(¯¯¯¯¯¯¯¯¯z2z3×z2z3)=4(|z2|×|z3|)2=4 (|z2|=|z3|=1)

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