Let a and b ∈R such that 0<a<1,0<b<1. The values of a and b such that the complex numbers z1=-a+

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Let a and b ∈R such that 0<a<1,0<b<1. The values of a and b...

Let a and $\mathrm{b} \in \mathrm{R}$ such that $0<\mathrm{a}<1,0<\mathrm{b}<1$. The values of $\mathrm{a}$ and $\mathrm{b}$ such that the complex numbers $\mathrm{z}_{1}=-\mathrm{a}+\mathrm{i}, \mathrm{z}_{2}=-1$ $+\mathrm{bi}$ and $\mathrm{z}_{\mathrm{a}}=0$ form an equilateral triangle are (a) $a=b=2-\sqrt{3}$ (b) $a=2-\sqrt{3}, b=2+\sqrt{3}$ (c) $a=\sqrt{3}, b=-\sqrt{3}$ (d) $2,2 \sqrt{3}$

Let a and $\mathrm{b} \in \mathrm{R}$ such that $0<\mathrm{a}<1,0<\mathrm{b}<1$. The values of $\mathrm{a}$ and $\mathrm{b}$ such that the complex numbers $\mathrm{z}_{1}=-\mathrm{a}+\mathrm{i}, \mathrm{z}_{2}=-1$ $+\mathrm{bi}$ and $\mathrm{z}_{\mathrm{a}}=0$ form an equilateral triangle are
(a) $a=b=2-\sqrt{3}$
(b) $a=2-\sqrt{3}, b=2+\sqrt{3}$
(c) $a=\sqrt{3}, b=-\sqrt{3}$
(d) $2,2 \sqrt{3}$

Key Concepts

Equilateral Triangle Properties

An equilateral triangle is characterized by all sides being of equal length and all internal angles being 60 degrees. When vertices of an equilateral triangle are represented by complex numbers, the equality of distances between each pair and the angular relationships provide useful constraints to solve for unknown parameters.

Complex Multiplication as Rotation

Multiplication by a complex number with unit modulus corresponds to a rotation in the complex plane. This property, particularly with rotations by 60 degrees or 120 degrees, is crucial in problems involving equilateral triangles, as it can relate the positions of the triangle's vertices through rotational symmetry.

Complex Number Representation

Complex numbers combine a real part and an imaginary part, naturally representing points in the two-dimensional plane. This algebraic framework permits the manipulation of geometric figures through operations such as addition, subtraction, and multiplication, facilitating the analysis of their properties.

Complex Plane Geometry

In the complex plane, geometric problems can be addressed using complex number arithmetic. Distances, angles, and geometric transformations, such as rotations and reflections, are efficiently handled using the modulus and argument of complex numbers.

Distance Formula for Complex Numbers

The distance between two complex numbers is given by the modulus of their difference. This formula is directly analogous to the Euclidean distance between two points in the plane and is fundamental in establishing the equality of sides in geometric figures like triangles.

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