The complex plane is given an orthonormal coordinate system ( $\mathrm { O } , \vec { u } , \vec { v }$ ). To every point $M$ with affixe $z$ in the plane, we associate the point $M ^ { \prime }$ with affixe $z ^ { \prime }$ defined by: $$z ^ { \prime } = z ^ { 2 } + 4 z + 3 .$$
A point $M$ is called invariant when it coincides with the associated point $M ^ { \prime }$. Prove that there exist two invariant points. Give the affixe of each of these points in algebraic form, then in exponential form.
Let A be the point with affixe $\frac { - 3 - \mathrm { i } \sqrt { 3 } } { 2 }$ and B the point with affixe $\frac { - 3 + \mathrm { i } \sqrt { 3 } } { 2 }$. Show that OAB is an equilateral triangle.
Determine the set $\mathcal { E }$ of points $M$ with affixe $z = x + \mathrm { i } y$ where $x$ and $y$ are real, such that the associated point $M ^ { \prime }$ lies on the real axis.
In the complex plane, represent the points A and B as well as the set $\mathcal { E }$.
The complex plane is given an orthonormal coordinate system ( $\mathrm { O } , \vec { u } , \vec { v }$ ). To every point $M$ with affixe $z$ in the plane, we associate the point $M ^ { \prime }$ with affixe $z ^ { \prime }$ defined by:
$$z ^ { \prime } = z ^ { 2 } + 4 z + 3 .$$
\begin{enumerate}
\item A point $M$ is called invariant when it coincides with the associated point $M ^ { \prime }$.
Prove that there exist two invariant points. Give the affixe of each of these points in algebraic form, then in exponential form.
\item Let A be the point with affixe $\frac { - 3 - \mathrm { i } \sqrt { 3 } } { 2 }$ and B the point with affixe $\frac { - 3 + \mathrm { i } \sqrt { 3 } } { 2 }$.
Show that OAB is an equilateral triangle.
\item Determine the set $\mathcal { E }$ of points $M$ with affixe $z = x + \mathrm { i } y$ where $x$ and $y$ are real, such that the associated point $M ^ { \prime }$ lies on the real axis.
\item In the complex plane, represent the points A and B as well as the set $\mathcal { E }$.
\end{enumerate}