The plane is equipped with a direct orthonormal coordinate system $(\mathrm{O}; \vec{u}, \vec{v})$.

The purpose of this exercise is to determine the non-zero complex numbers $z$ such that the points with affixes $1$, $z^2$ and $\dfrac{1}{z}$ are collinear.\\
On the graph provided in the appendix, point A has affix 1.

\textbf{Part A: study of examples}

\textbf{1. A first example}

In this question, we set $z = \mathrm{i}$.\\
a. Give the algebraic form of the complex numbers $z^2$ and $\dfrac{1}{z}$.\\
b. Plot the points $N_1$ with affix $z^2$, and $P_1$ with affix $\dfrac{1}{z}$ on the graph given in the appendix. We note that in this case the points $\mathrm{A}$, $N_1$ and $P_1$ are not collinear.

\textbf{2. An equation}

Solve in the set of complex numbers the equation with unknown $z$: $z^2 + z + 1 = 0$.

\textbf{3. A second example}

In this question, we set: $z = -\dfrac{1}{2} + \mathrm{i}\dfrac{\sqrt{3}}{2}$.\\
a. Determine the exponential form of $z$, then those of the complex numbers $z^2$ and $\dfrac{1}{z}$.\\
b. Plot the points $N_2$ with affix $z^2$ and $P_2$, with affix $\dfrac{1}{z}$ on the graph given in the appendix. We note that in this case the points $\mathrm{A}$, $N_2$ and $P_2$ are collinear.

\textbf{Part B}

Let $z$ be a non-zero complex number.\\
We denote by $N$ the point with affix $z^2$ and $P$ the point with affix $\dfrac{1}{z}$.

\begin{enumerate}
  \item Establish that, for every complex number different from 0, we have:
$$z^2 - \frac{1}{z} = \left(z^2 + z + 1\right)\left(1 - \frac{1}{z}\right)$$
  \item We recall that if $\vec{U}$ is a non-zero vector and $\vec{V}$ is a vector with affixes respectively $z_{\vec{U}}$ and $z_{\vec{V}}$, the vectors $\vec{U}$ and $\vec{V}$ are collinear if and only if there exists a real number $k$ such that $z_{\vec{V}} = k z_{\vec{U}}$.\\
Deduce that, for $z \neq 0$, the points $\mathrm{A}$, $N$ and $P$ defined above are collinear if and only if $z^2 + z + 1$ is a real number.
  \item We set $z = x + \mathrm{i}y$, where $x$ and $y$ denote real numbers.\\
Justify that: $z^2 + z + 1 = x^2 - y^2 + x + 1 + \mathrm{i}(2xy + y)$.
  \item a. Determine the set of points $M$ with affix $z \neq 0$ such that the points $\mathrm{A}$, $N$ and $P$ are collinear.\\
b. Trace this set of points on the graph given in the appendix.
\end{enumerate}