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If $z_{1}$ and $z_{2}$ are the roots of the equation $z^{2} + a z + b = 0,$ if the origin, $z_{1}$ and $z_{2}$ form an equilateral triangle then

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$a^{2} = 3 b$

$a^{2} + 3 b = 0$

$b^{2} = 3 a$

$b^{2} + 3 a = 0$

answer is A.

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Detailed Solution

$z_{1} + z_{2} = - a, z_{1} z_{2} = b$

Now, $0, z_{1}, z_{2}$ form an equilateral triangle

$\begin{array}{lc} \Leftrightarrow & 0^{2} + z_{1}^{2} + z_{2}^{2} = 0 (z_{1} + z_{2}) + z_{1} z_{2} \\ \Leftrightarrow & {(z_{1} + z_{2})}^{2} = 3 z_{1} z_{2} \\ \Leftrightarrow & a^{2} = 3 b \end{array}$

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