Suppose that z1,z2,z3 are three vertices of an equilateral triangle in the Argand plane. Let α =(1/2)(√3 +i) abd β be a non-zero complex number. The points α z1+β,α z2+β,α z3+β will be

Question

Suppose that z1,z2,z3 are three vertices of an equilateral triangle in the Argand plane. Let α =(1/2)(√3 +i) abd β be a non-zero complex number. The points α z1+β,α z2+β,α z3+β will be (A) The vertices of an equilateral triangle (B) The vertices of an isosceles triangle (C) Collinear (D) The vertices of an scalene triangle

Tardigrade Admin · Accepted Answer

(1/(α z1+β)-(α z2+β))+(1/(α z3+β)-(α z3+β)) =(1/α(z1-z2))+(1/α(z2-z3))+(1/α(z3-z1)) =(1/α)((1/z1-z2)+(1/z2-z3)+(1/z3-z1)) =(1/α)(0)=0 [∵ z1, z2, z3 are three vertices of an equilateral triangle] ∴ α z1+β, α z2+β, α z3+β are vertices of an equilateral triangle

Q. Suppose that z1​,z2​,z3​ are three vertices of an equilateral triangle in the Argand plane. Let α=21​(3​+i) abd β be a non-zero complex number. The points αz1​+β,αz2​+β,αz3​+β will be

The vertices of an equilateral triangle

The vertices of an isosceles triangle

Collinear

The vertices of an scalene triangle

Q. Suppose that $z_{1}, z_{2}, z_{3}$ are three vertices of an equilateral triangle in the Argand plane.
Let $α = \frac{1}{2} (3 + i)$ abd $β$ be a non-zero complex number. The points $α z_{1} + β, α z_{2} + β, α z_{3} + β$ will be