Space is referred to an orthonormal coordinate system $( \mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k } )$.
We consider the points $\mathrm { A } ( 1 ; 1 ; 4 ) ; \mathrm { B } ( 4 ; 2 ; 5 ) ; \mathrm { C } ( 3 ; 0 ; - 2 )$ and $\mathrm { J } ( 1 ; 4 ; 2 )$.
We denote:
\begin{itemize}
  \item $\mathscr { P }$ the plane passing through points $\mathrm { A }$, $\mathrm { B }$ and $\mathrm { C }$;
  \item $\mathscr { D }$ the line passing through point $\mathrm { J }$ and with direction vector $\overrightarrow { \mathrm { u } } \left( \begin{array} { l } 1 \\ 1 \\ 3 \end{array} \right)$.
\end{itemize}

\begin{enumerate}
  \item Relative position of $\mathscr { P }$ and $\mathscr { D }$\\
a. Show that the vector $\vec { n } \left( \begin{array} { c } 1 \\ - 4 \\ 1 \end{array} \right)$ is normal to $\mathscr { P }$.\\
b. Determine a Cartesian equation of the plane $\mathscr { P }$.\\
c. Show that $\mathscr { D }$ is parallel to $\mathscr { P }$.
\end{enumerate}

We consider the point $\mathrm { I } ( 1 ; 9 ; 0 )$ and we call $\mathscr { S }$ the sphere with center $\mathrm { I }$ and radius 6.

\begin{enumerate}
  \setcounter{enumi}{1}
  \item Relative position of $\mathscr { P }$ and $\mathscr { S }$\\
a. Show that the line $\Delta$ passing through $\mathrm { I }$ and perpendicular to the plane $\mathscr { P }$ intersects this plane $\mathscr { P }$ at the point $\mathrm { H } ( 3 ; 1 ; 2 )$.\\
b. Calculate the distance $\mathrm { IH }$.\\
We admit that for every point $M$ of the plane $\mathscr { P }$ we have $\mathrm { I } M \geqslant \mathrm { IH }$.\\
c. Does the plane $\mathscr { P }$ intersect the sphere $\mathscr { S }$? Justify your answer.
  \item Relative position of $\mathscr { D }$ and $\mathscr { S }$\\
a. Determine a parametric representation of the line $\mathscr { D }$.\\
b. Show that a point $M$ with coordinates $( x ; y ; z )$ belongs to the sphere $\mathscr { S }$ if and only if:
$$( x - 1 ) ^ { 2 } + ( y - 9 ) ^ { 2 } + z ^ { 2 } = 36 .$$
c. Show that the line $\mathscr { D }$ intersects the sphere at two distinct points.
\end{enumerate}