Suppose z1, z2, z3 are the vertices of an equilateral triangle inscribed in the circle | z| = 2. If z1 = √ 3i and z1,z2,z3 are in . the clockwise sense, then

Question

Suppose z1, z2, z3 are the vertices of an equilateral triangle inscribed in the circle | z| = 2. If z1 = √ 3i and z1,z2,z3 are in . the clockwise sense, then (A) z2 = 1 - √ 3i,z3 =-2 (B) z2 = , z3 = 1 - √ 3i (C) z2 = -1 + √ 3i,z3 =-2 (D) none of these

Tardigrade Admin · Accepted Answer

amp. (z1)=amp. (1+√3i) <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/91ddd424a013cb18e8e0a62375361389-.jpeg" /> =tan-1 √3=(π/3) amp. (z2)=(π/3)-(2π/3)=-(π/3) and |z2|=2 ∴ z2=2[cos(-(π/3))+i sin (-(π/3))] =2[cos (π/3)-i sin (π/3)] =2((1/2)-i (√3/2))=1-i√3 amp. (z3)=(π/3)+(2π/3)=π and |z3|=2 ∴ z3=2(cos π+i sin π)=2

Q. Suppose $z_{1}, z_{2}, z_{3}$ are the vertices of an equilateral triangle inscribed in the circle $∣ z ∣ = 2$ . If $z_{1}$ = $3 i$ and $z_{1}, z_{2}, z_{3}$ are in . the clockwise sense, then

$z_{2} = 1 - 3 i, z_{3} = - 2$

$z_{2} =, z_{3} = 1 - 3 i$

$z_{2} = - 1 + 3 i, z_{3} = - 2$

none of these

Q. Suppose z1​,z2​,z3​ are the vertices of an equilateral triangle inscribed in the circle ∣z∣=2. If z1​ = 3i​ and z1​,z2​,z3​ are in . the clockwise sense, then

z2​=1−3i​,z3​=−2

z2​=,z3​=1−3i​

z2​=−1+3i​,z3​=−2

none of these

amp.(z1​)=amp.(1+3​i) =tan−13​=3π​ amp.(z2​)=3π​−32π​=−3π​ and ∣z2​∣=2 ∴z2​=2[cos(−3π​)+isin(−3π​)] =2[cos3π​−isin3π​] =2(21​−i23​​)=1−i3​ amp.(z3​)=3π​+32π​=π and ∣z3​∣=2 ∴z3​=2(cosπ+isinπ)=2

Q. Suppose $z_{1}, z_{2}, z_{3}$ are the vertices of an equilateral triangle inscribed in the circle $∣ z ∣ = 2$ . If $z_{1}$ = $3 i$ and $z_{1}, z_{2}, z_{3}$ are in . the clockwise sense, then

$z_{2} = 1 - 3 i, z_{3} = - 2$

$z_{2} =, z_{3} = 1 - 3 i$

$z_{2} = - 1 + 3 i, z_{3} = - 2$