\textbf{Exercise 4 (5 points) -- Candidates who have chosen the specialization option}

A software allows transforming a rectangular element of a photograph. Thus, the initial rectangle OEFG is transformed into a rectangle $OE'F'G'$, called the image of OEFG.

The purpose of this exercise is to study the rectangle obtained after several successive transformations.

\textbf{Part A}

The plane is referred to an orthonormal coordinate system $(O, \vec{\imath}, \vec{\jmath})$.\\
The points E, F and G have coordinates respectively $(2; 2)$, $(-1; 5)$ and $(-3; 3)$.\\
The software transformation associates with any point $M(x; y)$ of the plane the point $M'(x'; y')$, image of point $M$ such that:
$$\left\{\begin{aligned}
x' &= \frac{5}{4}x + \frac{3}{4}y \\
y' &= \frac{3}{4}x + \frac{5}{4}y
\end{aligned}\right.$$

\begin{enumerate}
  \item a. Calculate the coordinates of points $E'$, $F'$ and $G'$, images of points E, F and G by this transformation.\\
b. Compare the lengths OE and $OE'$ on one hand, OG and $OG'$ on the other hand.
\end{enumerate}

Give the square matrix of order 2, denoted $A$, such that: $\binom{x'}{y'} = A\binom{x}{y}$.

\textbf{Part B}

In this part, we study the coordinates of the successive images of vertex F of rectangle OEFG when the software transformation is applied multiple times.

\begin{enumerate}
  \item Consider the algorithm intended to display the coordinates of these successive images.
  \item a. Prove that, for every natural integer $n$, the point $E_{n}$ is located on the line with equation $y = x$.\\
One may use the fact that, for every natural integer $n$, the coordinates $(x_{n}; y_{n})$ of point $E_{n}$ satisfy:
$$\binom{x_{n}}{y_{n}} = A^{n}\binom{2}{2}$$
b. Prove that the length $\mathrm{O}E_{n}$ tends towards $+\infty$ when $n$ tends towards $+\infty$.
\end{enumerate}