Let z1 be a fixed point on the circle of radius 1 centered at the origin in the Argand plane and z1 ≠ ± 1. Consider an equilateral triangle inscribed in the circle with z1, z2, z3 as the vertices taken in the counter clockwise direction. Then z1z2z3 is equal to

Question

Let z1 be a fixed point on the circle of radius 1 centered at the origin in the Argand plane and z1 ≠ ± 1. Consider an equilateral triangle inscribed in the circle with z1, z2, z3 as the vertices taken in the counter clockwise direction. Then z1z2z3 is equal to (A) z21 (B) z31 (C) z41 (D) z1

Tardigrade Admin · Accepted Answer

Given, z1 ≠ ± 1 Since, z2 and z3 can be obtained by rotating vector representing through (2 π/3) and (4 π/3), respectively ∴ z2=z1 ω and z3=z1 ω2 ∴ z1 z2 z3 =z1 × z1 ω × z1 ω2 =z13 ω3 =z13 [∵ ω3=1]

$z_{1}^{2}$

$z_{1}^{3}$

$z_{1}^{4}$

$z_{1}$

Q. Let z1​ be a fixed point on the circle of radius 1 centered at the origin in the Argand plane and z1​=±1. Consider an equilateral triangle inscribed in the circle with z1​,z2​,z3​ as the vertices taken in the counter clockwise direction. Then z1​z2​z3​ is equal to

z12​

z13​

z14​

z1​

Given, z1​=±1 Since, z2​ and z3​ can be obtained by rotating vector representing through 32π​ and 34π​, respectively ∴z2​=z1​ω and z3​=z1​ω2 ∴z1​z2​z3​=z1​×z1​ω×z1​ω2 =z13​ω3 =z13​ [∵ω3=1]

$z_{1}^{2}$

$z_{1}^{3}$

$z_{1}^{4}$

$z_{1}$