Exercise 3 — Part B
In an orthonormal coordinate system of space, consider the point $\mathrm{A}(3; 1; -5)$ and the line $d$ with parametric representation $\left\{\begin{array}{rl} x &= 2t + 1 \\ y &= -2t + 9 \\ z &= t - 3 \end{array}\right.$ where $t \in \mathbb{R}$.
1. Determine a Cartesian equation of the plane $P$ orthogonal to the line $d$ and passing through point A.
2. Show that the intersection point of plane $P$ and line $d$ is point $\mathrm{B}(5; 5; -1)$.
3. Justify that point $\mathrm{C}(7; 3; -9)$ belongs to plane $P$ then show that triangle ABC is a right isosceles triangle at A.
4. Let $t$ be a real number different from 2 and $M$ the point with parameter $t$ belonging to line $d$.
a. Justify that triangle $\mathrm{AB}M$ is right-angled.
b. Show that triangle $\mathrm{AB}M$ is isosceles at B if and only if the real number $t$ satisfies the equation $t^2 - 4t = 0$.
c. Deduce the coordinates of points $M_1$ and $M_2$ on line $d$ such that the right triangles $\mathrm{AB}M_1$ and $\mathrm{AB}M_2$ are isosceles at B.