Multiple choice

Let $z_1, z_2$ be two roots of the equation $z^2 + az + b = 0$, $z $ being a complex number. Further assume that the origin, $z_1$ and $z_2$ form an equilateral triangle. Then,

  1. $a^2 = b$
  2. $a^2 = 2b$
  3. $a^2 = 3b$
  4. $a^2 = 4b$
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C Correct answer
Explanation

For the origin, z1, and z2 to form an equilateral triangle, the condition is z1^2 + z2^2 - z1*z2 = 0. Since z1 and z2 are roots of z^2 + az + b = 0, we have z1 + z2 = -a and z1*z2 = b. Substituting (z1+z2)^2 - 3*z1*z2 = 0, we get (-a)^2 - 3*b = 0, so a^2 = 3b.

AI explanation

For an equilateral triangle with one vertex at the origin, rotating vector z1 by 60 degrees gives vector z2. This relation implies z2 = z1 * e^(i*pi/3), and substituting this into the sum z1 + z2 = -a yields a magnitude squared of 3 times the product b. Therefore, the correct identity is a^2 = 3b.

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