Let ABCDEFGH be a cube. We consider:
\begin{itemize}
  \item I and J the midpoints respectively of segments [AD] and [BC];
  \item P the center of face ABFE, that is, the intersection of diagonals (AF) and (BE);
  \item Q the midpoint of segment [FG].
\end{itemize}

We place ourselves in the orthonormal coordinate system $( \mathrm { A } ; \frac { 1 } { 2 } \overrightarrow { \mathrm { AB } } , \frac { 1 } { 2 } \overrightarrow { \mathrm { AD } } , \frac { 1 } { 2 } \overrightarrow { \mathrm { AE } } )$.\\
Throughout the exercise, we may use the coordinates of the points in the figure without justifying them.\\
We admit that a parametric representation of the line (IJ) is

$$\left\{ \begin{array} { l } 
x = r \\
y = 1 , \quad r \in \mathbf { R } \\
z = 0
\end{array} \right.$$

\begin{enumerate}
  \item Verify that a parametric representation of the line (PQ) is
\end{enumerate}

$$\left\{ \begin{array} { l } 
x = 1 + t \\
y = t , \quad t \in \mathbf { R } \\
z = 1 + t
\end{array} \right.$$

Let $t$ be a real number and $\mathrm { M } ( 1 + t ; t ; 1 + t )$ be the point on the line (PQ) with parameter $t$.\\
2. a. We admit that there exists a unique point K belonging to the line (IJ) such that (MK) is orthogonal to (IJ).\\
Prove that the coordinates of this point $\mathrm { K }$ are $( 1 + t ; 1 ; 0 )$.\\
b. Deduce that $\mathrm { MK } = \sqrt { 2 + 2 t ^ { 2 } }$.\\
3. a. Verify that $y - z = 0$ is a Cartesian equation of the plane (HGB).\\
b. We admit that there exists a unique point L belonging to the plane (HGB) such that (ML) is orthogonal to (HGB).\\
Verify that the coordinates of this point L are $\left( 1 + t ; \frac { 1 } { 2 } + t ; \frac { 1 } { 2 } + t \right)$.\\
c. Deduce that the distance ML is independent of $t$.\\
4. Does there exist a value of $t$ for which the distance MK is equal to the distance ML?