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 i≠j≠k 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+z2z3∈R ∴z1=¯¯¯¯¯¯¯¯¯z2z3⇒reiθ1=re−iθ2⋅re−iθ3⇒r=r2e−i(θ1+θ2+θ3)⇒re−i(θ1+θ2+θ3)=1 (∵r≠0) 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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