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Let \( z_{1} \) and \( z_{2} \) be the roots of the equation \( z^{2}+p z+q=0 \), where the coefficients \( p \) and \( q \) may be complex numbers.
\( \mathrm{P} \) Let \( A \) and \( B \) represent \( z_{1} \) and \( z_{2} \) in the complex plane. If

W \( \angle A O B=\alpha \neq 0 \) and \( O A=O B \), where \( O \) is the origin prove that \( p^{2}=4 q \cos ^{2}\left(\frac{\alpha}{2}\right) \).
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