Suppose that z1, z2, z3 are three vertices of an equilateral triangle in the Argand plane. Let α=(1/2)(√3+i) and 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) and 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

Since, z1, z2 and z3 are the vertices of an equilateral triangle, therefore |z1-z2| =|z2-z3| =|z3-z1|=k (say) Also, α=(1/2)(√3+i) ⇒ |α| =(1/2) √3+1=(1/2) × 2=1 Let A =α z1+β, B=α z2+β and C=α z3+β Now, |A B|=|α z2+β-(α z1+β)| =|α(z2-z1)| =|α||z2-z1| =|1||z2-z1| =1|z2-z1| =|z2-z1|=k Similarly, B C=C A=k Hence, the points α z1+β, α z2+β and α z3+β are the 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) and 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 $α = \frac{1}{2} (3 + i)$ and $β$ be a non-zero complex number. The points $α z_{1} + β, α z_{2} + β, α z_{3} + β$ will be