Let $a$ be a strictly positive real number.\\
We consider the function $f$ defined on the interval $]0; +\infty[$ by
$$f(x) = a\ln(x)$$
We denote $\mathscr{C}$ its representative curve in an orthonormal coordinate system.\\
Let $x_0$ be a real number strictly greater than 1.

\begin{enumerate}
  \item Determine the abscissa of the point of intersection of the curve $\mathscr{C}$ and the x-axis.
  \item Verify that the function $F$ defined by $F(x) = a[x\ln(x) - x]$ is a primitive of the function $f$ on the interval $]0; +\infty[$.
  \item Deduce the area of the blue region as a function of $a$ and $x_0$.
\end{enumerate}

We denote $T$ the tangent line to the curve $\mathscr{C}$ at the point $M$ with abscissa $x_0$.\\
We call $A$ the point of intersection of the tangent line $T$ with the y-axis and $B$ the orthogonal projection of $M$ onto the y-axis.

\begin{enumerate}
  \setcounter{enumi}{3}
  \item Prove that the length AB is equal to a constant (that is, to a number that does not depend on $x_0$) which we will determine.\\
  The candidate will take care to make their approach explicit.
\end{enumerate}