Exercise 4 (3 points) In the complex plane equipped with a direct orthonormal coordinate system $(\mathrm{O}; \vec{u}, \vec{v})$, we consider the points A and B with complex numbers respectively $z_{\mathrm{A}} = 2\mathrm{e}^{\mathrm{i}\frac{\pi}{4}}$ and $z_{\mathrm{B}} = 2\mathrm{e}^{\mathrm{i}\frac{3\pi}{4}}$.
Show that OAB is a right isosceles triangle.
We consider the equation $$(E): z^2 - \sqrt{6}\, z + 2 = 0$$ Show that one of the solutions of $(E)$ is the complex number of a point located on the circumscribed circle of triangle OAB.
\textbf{Exercise 4} (3 points)
In the complex plane equipped with a direct orthonormal coordinate system $(\mathrm{O}; \vec{u}, \vec{v})$, we consider the points A and B with complex numbers respectively $z_{\mathrm{A}} = 2\mathrm{e}^{\mathrm{i}\frac{\pi}{4}}$ and $z_{\mathrm{B}} = 2\mathrm{e}^{\mathrm{i}\frac{3\pi}{4}}$.
\begin{enumerate}
\item Show that OAB is a right isosceles triangle.
\item We consider the equation
$$(E): z^2 - \sqrt{6}\, z + 2 = 0$$
Show that one of the solutions of $(E)$ is the complex number of a point located on the circumscribed circle of triangle OAB.
\end{enumerate}