Question

Let \(z_{1}\) and \(z_{2}\) be two imaginary roots of \(z^{2}+p z+q=0\), where \(p\) and \(q\) are real. The points \(z_{1}, z_{2}\) and origin form an equilateral triangle if
(A) \(p^{2}>3 q\)
(B) \(p^{2}<3 q\)
(C) \(p^{2}=3 q\)
(D) \(p^{2}=q\)

Answer 1

Ans: (C)
Hint: \(\mathrm{O}^{2}+\mathrm{z}_{1}{ }^{2}+\mathrm{z}_{2}{ }^{2}=\mathrm{z}_{1} \mathrm{z}_{2}\)
$$
\begin{aligned}
&\Rightarrow z_{1}{ }^{2}+z_{2}{ }^{2}=z_{1} z_{2} \\
&\Rightarrow\left(z_{1}+z_{2}\right)^{2}=3 z_{1} z_{2} \\
&\Rightarrow p^{2}=3 q
\end{aligned}
$$

Answer 2

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