
On a:
\begin{align*}
z^2 + \overline{z}^2 = 1 & \iff (a+\i b)^2 + (a-\i b)^2 = 1\\
& \iff a^2 + 2ab\i + (\i b)^2 + a^2 - 2ab\i + (\i b)^2 = 1\\
& \iff a^2 + 2ab\i - b^2 + a^2 - 2ab\i - b^2 = 1\\
& \iff 2a^2 - 2b^2 = 1\\
& \iff a^2 - b^2 = \dfrac{1}{2}\\
& \iff b^2 = a^2 - \dfrac{1}{2}.
\end{align*}
