Let z1, z2, z3 be three vertices of an equilateral triangle circumscribing the circle |z|=(1/2) . If z1=(1/2)+(i √3/2) and z1, z2 z3 were in anticlockwise sense, then z2 is

Question

Let z1, z2, z3 be three vertices of an equilateral triangle circumscribing the circle |z|=(1/2) . If z1=(1/2)+(i √3/2) and z1, z2 z3 were in anticlockwise sense, then z2 is (A) 1+√3 i (B) 1-√3 i (C) 1 (D) -1

Tardigrade Admin · Accepted Answer

Given that, |z|=(1/2) and z1=(1/2)+(i √3/2) Here, one of the number must be a conjugate of z1=(1/2)+(i √3/2) i.e. z2=(1/2)-(i √3/2) and z2=z1 e-i 2 π / 3 ⇒ z2=((1/2)+(i √3/2))[ cos (2 π/3)+i sin (2 π/3)] =-2 × (1/2)=-1 .[∵(1+i √3) cos (2 π/3)+i sin (2 π/3))=-2]

Q. Let $z_{1}, z_{2}, z_{3}$ be three vertices of an equilateral triangle circumscribing the circle $∣ z ∣ = \frac{1}{2} .$ If $z_{1} = \frac{1}{2} + \frac{i 3}{2}$ and $z_{1}, z_{2} z_{3}$ were in anticlockwise sense, then $z_{2}$ is

$1 + 3 i$

$1 - 3 i$

$1$

$- 1$

Q. Let z1​,z2​,z3​ be three vertices of an equilateral triangle circumscribing the circle ∣z∣=21​. If z1​=21​+2i3​​ and z1​,z2​z3​ were in anticlockwise sense, then z2​ is

1+3​i

1−3​i

1

−1

Given that, ∣z∣=21​ and z1​=21​+2i3​​ Here, one of the number must be a conjugate of z1​=21​+2i3​​ i.e. z2​=21​−2i3​​ and z2​=z1​e−i2π/3 ⇒z2​=(21​+2i3​​)[cos32π​+isin32π​] =−2×21​=−1 [∵(1+i3​)cos32π​+isin32π​)=−2]

Q. Let $z_{1}, z_{2}, z_{3}$ be three vertices of an equilateral triangle circumscribing the circle $∣ z ∣ = \frac{1}{2} .$ If $z_{1} = \frac{1}{2} + \frac{i 3}{2}$ and $z_{1}, z_{2} z_{3}$ were in anticlockwise sense, then $z_{2}$ is

$1 + 3 i$

$1 - 3 i$

$1$

$- 1$