In this exercise, $x$ and $y$ are real numbers greater than 1.\\
In the complex plane equipped with a direct orthonormal coordinate system ( $\mathrm { O } ; \vec { u } , \vec { v }$ ), we consider the points $\mathrm { A} , \mathrm { B }$ and C with affixes respectively $z_{\mathrm{A}} = 1 + \mathrm{i}$, $z_{\mathrm{B}} = x + \mathrm{i}$, $z_{\mathrm{C}} = y + \mathrm{i}$.

Problem: we seek the possible values of real numbers $x$ and $y$, greater than 1, for which :

$$\mathrm { OC } = \mathrm { OA } \times \mathrm { OB } \quad \text { and } ( \vec { u } , \overrightarrow { \mathrm { OB } } ) + ( \vec { u } , \overrightarrow { \mathrm { OC } } ) = ( \vec { u } , \overrightarrow { \mathrm { OA } } )$$

\begin{enumerate}
  \item Prove that if $\mathrm { OC } = \mathrm { OA } \times \mathrm { OB }$, then $y ^ { 2 } = 2 x ^ { 2 } + 1$.
  \item Reproduce on your answer sheet and complete the following algorithm so that it displays all couples $( x , y )$ such that :
\begin{verbatim}
For x going from 1 to ... do
    For...
        If...
            Display x and y
        End If
    End For
End For
\end{verbatim}
When this algorithm is executed, it displays the value 2 for variable $x$ and the value 3 for variable $y$.
  \item Study of a particular case: in this question only, we take $x = 2$ and $y = 3$.\\
a. Give the modulus and an argument of $z _ { \mathrm { A } }$.\\
b. Show that $\mathrm { OC } = \mathrm { OA } \times \mathrm { OB }$.\\
c. Show that $z _ { \mathrm { B } } z _ { \mathrm { C } } = 5 z _ { \mathrm { A } }$ and deduce that $( \vec { u } , \overrightarrow { \mathrm { OB } } ) + ( \vec { u } , \overrightarrow { \mathrm { OC } } ) = ( \vec { u } , \overrightarrow { \mathrm { OA } } )$.
  \item We return to the general case, and we seek whether there exist other values of real numbers $x$ and $y$ such that points $\mathrm { A } , \mathrm { B }$ and C satisfy the two conditions :\\
$\mathrm { OC } = \mathrm { OA } \times \mathrm { OB }$ and $( \vec { u } , \overrightarrow { \mathrm { OB } } ) + ( \vec { u } , \overrightarrow { \mathrm { OC } } ) = ( \vec { u } , \overrightarrow { \mathrm { OA } } )$.\\
Recall that if $\mathrm { OC } = \mathrm { OA } \times \mathrm { OB }$, then $y ^ { 2 } = 2 x ^ { 2 } + 1$ (question 1.).\\
a. Prove that if $( \vec { u } , \overrightarrow { \mathrm { OB } } ) + ( \vec { u } , \overrightarrow { \mathrm { OC } } ) = ( \vec { u } , \overrightarrow { \mathrm { OA } } )$, then $\arg \left[ \frac { ( x + \mathrm { i } ) ( y + \mathrm { i } ) } { 1 + \mathrm { i } } \right] = 0 \bmod 2 \pi$.

Deduce that under this condition : $x + y - x y + 1 = 0$.\\
b. Prove that if the two conditions are satisfied and moreover $x \neq 1$, then :

$$y = \sqrt { 2 x ^ { 2 } + 1 } \quad \text { and } y = \frac { x + 1 } { x - 1 }$$

  \item We define the functions $f$ and $g$ on the interval $] 1 ; + \infty [$ by :

$$f ( x ) = \sqrt { 2 x ^ { 2 } + 1 } \quad \text { and } g ( x ) = \frac { x + 1 } { x - 1 }$$

Determine the number of solutions to the initial problem.\\
We may use the function $h$ defined on the interval $] 1 ; + \infty [$ by $h ( x ) = f ( x ) - g ( x )$ and rely on the screenshot of a computer algebra software given below.
\end{enumerate}