Theme: Exponential function Main topics covered: Geometry in space The space is equipped with an orthonormal coordinate system $( \mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k } )$. We consider the points $\mathrm { A } ( 5 ; 0 ; - 1 ) , \mathrm { B } ( 1 ; 4 ; - 1 ) , \mathrm { C } ( 1 ; 0 ; 3 ) , \mathrm { D } ( 5 ; 4 ; 3 )$ and $\mathrm { E } ( 10 ; 9 ; 8 )$.
a. Let R be the midpoint of the segment $[ \mathrm { AB } ]$. Calculate the coordinates of point R as well as the coordinates of the vector $\overrightarrow { \mathrm { AB } }$. b. Let $\mathscr { P } _ { 1 }$ be the plane passing through point R and for which $\overrightarrow { \mathrm { AB } }$ is a normal vector. Prove that a Cartesian equation of the plane $\mathscr { P } _ { 1 }$ is: $$x - y - 1 = 0 .$$ c. Prove that point E belongs to the plane $\mathscr { P } _ { 1 }$ and that $\mathrm { EA } = \mathrm { EB }$.
We consider the plane $\mathscr { P } _ { 2 }$ with Cartesian equation $x - z - 2 = 0$. a. Justify that the planes $\mathscr { P } _ { 1 }$ and $\mathscr { P } _ { 2 }$ are secant. b. We denote $\Delta$ the line of intersection of $\mathscr { P } _ { 1 }$ and $\mathscr { P } _ { 2 }$. Prove that a parametric representation of the line $\Delta$ is: $$\left\{ \begin{aligned}
x & = 2 + t \\
y & = 1 + t \quad ( t \in \mathbb { R } ) . \\
z & = t
\end{aligned} \right.$$
We consider the plane $\mathscr { P } _ { 3 }$ with Cartesian equation $y + z - 3 = 0$. Justify that the line $\Delta$ is secant to the plane $\mathscr { P } _ { 3 }$ at a point $\Omega$ whose coordinates you will determine. If S and T are two distinct points in space, we recall that the set of points M in space such that $\mathrm{MS} = \mathrm{MT}$ is a plane, called the perpendicular bisector plane of the segment $[ \mathrm { ST } ]$. We assume that the planes $\mathscr { P } _ { 1 }$, $\mathscr { P } _ { 2 }$ and $\mathscr { P } _ { 3 }$ are the perpendicular bisector planes of the segments [AB], [AC] and [AD] respectively.
a. Justify that $\Omega A = \Omega B = \Omega C = \Omega D$. b. Deduce that the points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D belong to the same sphere, whose centre and radius you will specify.
\section*{Exercise 4 — 6 points}
Theme: Exponential function\\
Main topics covered: Geometry in space\\
The space is equipped with an orthonormal coordinate system $( \mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k } )$.\\
We consider the points $\mathrm { A } ( 5 ; 0 ; - 1 ) , \mathrm { B } ( 1 ; 4 ; - 1 ) , \mathrm { C } ( 1 ; 0 ; 3 ) , \mathrm { D } ( 5 ; 4 ; 3 )$ and $\mathrm { E } ( 10 ; 9 ; 8 )$.
\begin{enumerate}
\item a. Let R be the midpoint of the segment $[ \mathrm { AB } ]$. Calculate the coordinates of point R as well as the coordinates of the vector $\overrightarrow { \mathrm { AB } }$.\\
b. Let $\mathscr { P } _ { 1 }$ be the plane passing through point R and for which $\overrightarrow { \mathrm { AB } }$ is a normal vector. Prove that a Cartesian equation of the plane $\mathscr { P } _ { 1 }$ is:
$$x - y - 1 = 0 .$$
c. Prove that point E belongs to the plane $\mathscr { P } _ { 1 }$ and that $\mathrm { EA } = \mathrm { EB }$.
\item We consider the plane $\mathscr { P } _ { 2 }$ with Cartesian equation $x - z - 2 = 0$.\\
a. Justify that the planes $\mathscr { P } _ { 1 }$ and $\mathscr { P } _ { 2 }$ are secant.\\
b. We denote $\Delta$ the line of intersection of $\mathscr { P } _ { 1 }$ and $\mathscr { P } _ { 2 }$. Prove that a parametric representation of the line $\Delta$ is:
$$\left\{ \begin{aligned}
x & = 2 + t \\
y & = 1 + t \quad ( t \in \mathbb { R } ) . \\
z & = t
\end{aligned} \right.$$
\item We consider the plane $\mathscr { P } _ { 3 }$ with Cartesian equation $y + z - 3 = 0$. Justify that the line $\Delta$ is secant to the plane $\mathscr { P } _ { 3 }$ at a point $\Omega$ whose coordinates you will determine.
If S and T are two distinct points in space, we recall that the set of points M in space such that $\mathrm{MS} = \mathrm{MT}$ is a plane, called the perpendicular bisector plane of the segment $[ \mathrm { ST } ]$. We assume that the planes $\mathscr { P } _ { 1 }$, $\mathscr { P } _ { 2 }$ and $\mathscr { P } _ { 3 }$ are the perpendicular bisector planes of the segments [AB], [AC] and [AD] respectively.
\item a. Justify that $\Omega A = \Omega B = \Omega C = \Omega D$.\\
b. Deduce that the points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D belong to the same sphere, whose centre and radius you will specify.
\end{enumerate}