Space is equipped with an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ).\\
We consider the points

$$\mathrm { A } ( 2 \sqrt { 3 } ; 0 ; 0 ) , \quad \mathrm { B } ( 0 ; 2 ; 0 ) , \quad \mathrm { C } ( 0 ; 0 ; 1 ) \quad \text { and } \quad \mathrm { K } \left( \frac { \sqrt { 3 } } { 2 } ; \frac { 3 } { 2 } ; 0 \right) .$$

\begin{enumerate}
  \item Justify that a parametric representation of the line (CK) is:
\end{enumerate}

$$\left\{ \begin{aligned}
x & = \frac { \sqrt { 3 } } { 2 } t \\
y & = \frac { 3 } { 2 } t \quad ( t \in \mathbb { R } ) \\
z & = 1 - t
\end{aligned} \right.$$

\begin{enumerate}
  \setcounter{enumi}{1}
  \item Let $\mathrm { M } ( t )$ be a point on the line (CK) parametrized by a real number $t$. Establish that $\mathrm { OM} ( t ) = \sqrt { 4 t ^ { 2 } - 2 t + 1 }$.
  \item Let $f$ be the function defined and differentiable on $\mathbb { R }$ by $f ( t ) = \mathrm { OM } ( t )$.\\
a. Study the variations of the function $f$ on $\mathbb { R }$.\\
b. Deduce the value of $t$ for which $f$ reaches its minimum.
  \item Deduce that the point $\mathrm { H } \left( \frac { \sqrt { 3 } } { 8 } ; \frac { 3 } { 8 } ; \frac { 3 } { 4 } \right)$ is the orthogonal projection of point O onto the line (CK).
  \item Prove, using the dot product tool, that point H is the orthocenter (intersection of the altitudes of a triangle) of triangle ABC.
  \item a. Prove that the line $( \mathrm { OH } )$ is orthogonal to the plane $( \mathrm { ABC } )$.\\
b. Deduce an equation of the plane (ABC).
  \item Calculate, in square units, the area of triangle ABC.
\end{enumerate}