Exercise 2
The complex plane is equipped with a direct orthonormal coordinate system $(\mathrm{O}; \vec{u}; \vec{v})$ with unit 2 cm. We call $f$ the function that, to any point $M$, distinct from point O and with affixe a complex number $z$, associates the point $M'$ with affixe $z'$ such that $$z' = -\frac{1}{z}$$
1. Consider the points A and B with affixes respectively $z_{\mathrm{A}} = -1 + \mathrm{i}$ and $z_{\mathrm{B}} = \frac{1}{2}\mathrm{e}^{\mathrm{i}\frac{\pi}{3}}$.
a. Determine the algebraic form of the affixe of point $\mathrm{A}'$ image of point A by the function $f$.
b. Determine the exponential form of the affixe of point $\mathrm{B}'$ image of point B by the function $f$.
c. On your paper, place the points $\mathrm{A}, \mathrm{B}, \mathrm{A}'$ and $\mathrm{B}'$ in the direct orthonormal coordinate system $(\mathrm{O}; \vec{u}; \vec{v})$. For points B and $\mathrm{B}'$, construction lines should be left visible.
2. Let $r$ be a strictly positive real number and $\theta$ a real number. Consider the complex number $z$ defined by $z = r\mathrm{e}^{\mathrm{i}\theta}$.
a. Show that $z' = \frac{1}{r}\mathrm{e}^{\mathrm{i}(\pi - \theta)}$.
b. Is it true that if a point $M$, distinct from O, belongs to the disk with center O and radius 1 without belonging to the circle with center O and radius 1, then its image $M'$ by the function $f$ is outside this disk? Justify.
3. Let the circle $\Gamma$ with center K with affixe $z_{\mathrm{K}} = -\frac{1}{2}$ and radius $\frac{1}{2}$.
a. Show that a Cartesian equation of the circle $\Gamma$ is $x^2 + x + y^2 = 0$.
b. Let $z = x + \mathrm{i}y$ with $x$ and $y$ not both zero. Determine the algebraic form of $z'$ as a function of $x$ and $y$.
c. Let $M$ be a point, distinct from O, on the circle $\Gamma$. Show that the image $M'$ of point $M$ by the function $f$ belongs to the line with equation $x = 1$.