Multiple choice

The roots $z_1$, $z_2$, $z_3$ of the equation $x^3 + 3ax^2 + 3bx + c = 0$ in which a, b, c are complex numbers, correspond to the points A,B, C on the complex plane. Then ABC is an equilateral triangle if

  1. $a^2=b$
  2. $a=b^2$
  3. $a+b^2=0$
  4. $a^2+b=0$
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A Correct answer
Explanation

For a cubic equation x^3 + 3ax^2 + 3bx + c = 0, the roots form an equilateral triangle in the complex plane if the derivative of the cubic has roots that are equal, or more specifically, if a^2 = b.

AI explanation

For the roots of the cubic to form an equilateral triangle in the complex plane, the centroid of the triangle must coincide with the point of inflection of the cubic curve. The x-coordinate of the centroid of the roots is the average of the roots, which equals -a. The point of inflection is found using the second derivative 6x + 6a = 0, which also yields x = -a. Translating the roots by substituting y = x + a transforms the cubic into the form y^3 + 3(b - a^2)y + (c - 3ab + 2a^3) = 0. For these translated roots to form an equilateral triangle centered at the origin, the sum of their y-coordinates must be zero and the sum of their squares must also be zero, which forces the coefficient of y to vanish. This requires 3(b - a^2) = 0, meaning a^2 = b.

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