Exercise 2 (4 points)

The complex plane is equipped with an orthonormal coordinate system $(\mathrm{O}, \vec{u}, \vec{v})$. To every point $M$ with affixe $z$, we associate the point $M'$ with affixe $$z' = -z^2 + 2z$$ The point $M'$ is called the image of point $M$.

1. Solve in the set $\mathbb{C}$ of complex numbers the equation: $$-z^2 + 2z - 2 = 0$$ Deduce the affixes of the points whose image is the point with affixe 2.
   
2. Let $M$ be a point with affixe $z$ and $M'$ its image with affixe $z'$.
   We denote $N$ the point with affixe $z_N = z^2$. Show that $M$ is the midpoint of segment $[NM']$.
   
3. In this question, we assume that the point $M$ with affixe $z$ belongs to the circle $\mathscr{C}$ with center O and radius 1. We denote $\theta$ an argument of $z$. a. Determine the modulus of each of the complex numbers $z$ and $z_N$, as well as an argument of $z_N$ as a function of $\theta$. b. On the figure given in the appendix on page 7, a point $M$ on the circle $\mathscr{C}$ has been represented. Construct on this figure the points $N$ and $M'$ using a ruler and compass (leave the construction lines visible). c. Let A be the point with affixe 1. What is the nature of triangle $AMM'$?

Consider the complex function $M(z) = \frac{mz}{mz - z + 1}$, where $m$ is a complex number such that $|m| = 1$ and $m \neq 1$.

1. Find all fixed points of $M(z)$ which satisfy $M(z) = z$.
2. Express the derivative of $M(z)$ at $z = 0$ by using $m$.
3. Find $m$ for which the circle $\left| z - \frac{1-i}{2} \right| = \frac{1}{\sqrt{2}}$ on the complex $z$ plane is mapped onto the real axis through $M(z)$.