Exercise 2 (For candidates who did not choose the mathematics speciality)

The complex plane is equipped with a direct orthonormal coordinate system $(\mathrm{O}, \vec{u}, \vec{v})$ (graphical unit: 4 cm). Let A be the point with affixe $z_{\mathrm{A}} = \mathrm{i}$ and B the point with affixe $z_{\mathrm{B}} = \mathrm{e}^{-\mathrm{i}\frac{5\pi}{6}}$.

\begin{enumerate}
  \item Let $r$ be the rotation with centre O and angle $\frac{2\pi}{3}$. Let C denote the image of B by $r$.\\
  a. Determine a complex expression for $r$.\\
  b. Show that the affixe of C is $z_{\mathrm{C}} = \mathrm{e}^{-\mathrm{i}\frac{\pi}{6}}$.\\
  c. Write $z_{\mathrm{B}}$ and $z_{\mathrm{C}}$ in algebraic form.\\
  d. Plot the points A, B and C.
  \item Let D be the centroid of points A, B and C with respective coefficients $2, -1$ and $2$.\\
  a. Show that the affixe of D is $z_{\mathrm{D}} = \frac{\sqrt{3}}{2} + \frac{1}{2}\mathrm{i}$. Plot point D.\\
  b. Show that A, B, C and D lie on the same circle.
  \item Let $h$ be the homothety with centre A and ratio 2. Let E denote the image of D by $h$.\\
  a. Determine a complex expression for $h$.\\
  b. Show that the affixe of E is $z_{\mathrm{E}} = \sqrt{3}$. Plot point E.
  \item a. Calculate the ratio $\frac{z_{\mathrm{D}} - z_{\mathrm{C}}}{z_{\mathrm{E}} - z_{\mathrm{C}}}$. Write the result in exponential form.\\
  b. Deduce the nature of triangle CDE.
\end{enumerate}