On an Argand diagram with origin $O$, the roots of this equation are shown by the two distinct points $A$ and $B$.
(a)[2]
Find the solutions of the equation $z^2 - 2piz - q = 0$, where $p$ and $q$ are real constants.
(b)[2]
Given that $A$ and $B$ lie on the imaginary axis, determine a relationship between $p$ and $q$.
(c)[3]
If triangle $OAB$ is equilateral instead, express $q$ in terms of $p$.
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