The roots $z_{1}, z_{2}, z_{3}$ of the equation $x^{3} + 3 a x^{2} + 3 b x + c = 0$ in which a, b, c are complex numbers, correspond to the points A, B, C on the Gaussian plane. Find the condition for triangle to be equilateral.

b^{2} = a

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a^{2} = 2 b

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a^{2} = b

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b^{2} = 2 a

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Solution

The correct option is C $a^{2} = b$ .
Since $z_{1}, z_{2}, z_{3}$ are the roots of $x^{3} + 3 a x^{2} + 3 b x + c = 0$ , we have
$z_{1} + z_{2} + z_{3} = - 3 a$ or $\frac{z_{1} + z_{2} + z_{3}}{3} = - a$
and $z_{1} z_{2} + z_{2} z_{3} + z_{3} z_{1} = 2 b$
Hence, the centroid of the triangle ABC is the point of affix $- a$ .
Now the triangle will be equilateral if
$z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = z_{1} z_{2} + z_{2} z_{3} + z_{3} z_{1}$
$\Rightarrow (z_{1} + z_{2} + z_{3})^{2} = 3 (z_{1} z_{2} + z_{2} z_{3} + z_{3} z_{1})$
$\Rightarrow (- 3 a)^{2} = 3 (3 b)$
$\Rightarrow a^{2} = b$
Ans: C

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