In space with an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ), we consider the points

$$\mathrm { A } ( 0 ; 4 ; 16 ) , \quad \mathrm { B } ( 0 ; 4 ; - 10 ) , \quad \mathrm { C } ( 4 ; - 8 ; 0 ) \quad \text { and } \quad \mathrm { K } ( 0 ; 4 ; 3 ) .$$

We define the sphere $S$ with center K and radius 13 as the set of points M such that $\mathrm { KM } = 13$.

\begin{enumerate}
  \item a. Verify that point C belongs to sphere $S$.\\
b. Show that triangle ABC is right-angled at C.
  \item a. Show that the vector $\vec { n } \left( \begin{array} { l } 3 \\ 1 \\ 0 \end{array} \right)$ is a normal vector to plane (ABC).\\
b. Determine a Cartesian equation of plane (ABC).
  \item We admit that sphere $S$ intersects the x-axis at two points, one having a positive abscissa and the other a negative abscissa. We denote D the one with positive abscissa.\\
a. Show that point D has coordinates $( 12 ; 0 ; 0 )$.\\
b. Give a parametric representation of the line $\Delta$ passing through D and perpendicular to plane (ABC).\\
c. Determine the distance from point D to plane (ABC).
  \item Calculate an approximate value, to the nearest unit of volume, of the volume of tetrahedron ABCD. We recall the formula for the volume V of a tetrahedron
$$V = \frac { 1 } { 3 } \times \mathscr { B } \times h$$
where $\mathscr { B }$ is the area of a base and h the associated height.
\end{enumerate}