We place ourselves in the complex plane with coordinate system $(O ; \vec { u } , \vec { v })$.\\
Let $f$ be the transformation that associates to any non-zero complex number $z$ the complex number $f ( z )$ defined by:

$$f ( z ) = z + \frac { 1 } { z }$$

We denote by $M$ the point with affixe $z$ and $M ^ { \prime }$ the point with affixe $f ( z )$.

\begin{enumerate}
  \item We call A the point with affixe $a = - \frac { \sqrt { 2 } } { 2 } + \mathrm { i } \frac { \sqrt { 2 } } { 2 }$.\\
a. Determine the exponential form of $a$.\\
b. Determine the algebraic form of $f ( a )$.
  \item Solve, in the set of complex numbers, the equation $f ( z ) = 1$.
  \item Let $M$ be a point with affixe $z$ on the circle $\mathscr { C }$ with center O and radius 1.\\
a. Justify that the affixe $z$ can be written in the form $z = \mathrm { e } ^ { \mathrm { i } \theta }$ with $\theta$ a real number.\\
b. Show that $f ( z )$ is a real number.
  \item Describe and represent the set of points $M$ with affixe $z$ such that $f ( z )$ is a real number.
\end{enumerate}