Let $P_1$ and $P_2$ be two planes given by $P_1: 10x + 15y + 12z - 60 = 0$, $P_2: -2x + 5y + 4z - 20 = 0$. Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on $P_1$ and $P_2$ ?

Let P₁ and P₂ be two planes given by P₁: 10x + 15y + 12z - 60 = 0, P₂: -2x + 5y + 4z - 20 = 0. Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on P₁ and P₂ ? $\text{Given: } P_1: 10x + 15y + 12z - 60 = 0$ $P_2: -2x + 5y + 4z - 20 = 0$ $\text{To find: a line that can be an edge of a tetrahedron with faces } P_1, P_2$ $(1)\ \dfrac{x-1}{0} = \dfrac{y-1}{0} = \dfrac{z-1}{5} \quad (2)\ \dfrac{x-6}{-5} = \dfrac{y}{2} = \dfrac{z}{3}$ $(3)\ \dfrac{x}{-2} = \dfrac{y-4}{5} = \dfrac{z}{4} \quad (4)\ \dfrac{x}{1} = \dfrac{y-4}{-2} = \dfrac{z}{3}$ $(0,4,0):\ P_1 \Rightarrow 15(4) - 60 = 0,\quad P_2 \Rightarrow 5(4) - 20 = 0$ $\therefore (0,4,0) \in L$ $\vec{n}_1 = (10,15,12),\quad \vec{n}_2 = (-2,5,4)$ $\vec{n}_1 \times \vec{n}_2 = (0,-64,80) \parallel (0,-4,5)$ $L: \vec{r} = (0,4,0) + \lambda(0,-4,5)$ $\text{Plan: } 1.\ \text{Classify the possible edges}$ $2.\ \text{Test each option against } L,\ P_1,\ P_2$ $\text{Faces } ABC \subset P_1,\ ABD \subset P_2 \Rightarrow AB \subset L$ $A, B \in L;\quad C \in P_1;\quad D \in P_2$ $AB \subset L$ $AC, BC \subset P_1;\quad AD, BD \subset P_2$ $CD:\ \text{meets } P_1, P_2 \text{ at two distinct points}$ $(1)\ \vec{d} = (0,0,5):\quad t_{P_1}=\dfrac{23}{60},\quad t_{P_2}=\dfrac{13}{20}$ $\text{distinct points, misses } L \Rightarrow \text{edge } CD \quad ✓$ $(2)\ \vec{d} = (-5,2,3):\ P_1 \text{ at } (6,0,0),\ P_2 \text{ at } (1,2,3)$ $\text{distinct points, misses } L \Rightarrow \text{edge } CD \quad ✓$ $(3)\ \vec{d} = (-2,5,4):\ P_1 \Rightarrow 103t = 0,\quad P_2 \Rightarrow 45t = 0$ $\Rightarrow \text{both planes met only at } t = 0 \Rightarrow (0,4,0) \in L$ $(3)\ \vec{d} = (-2,5,4) = \vec{n}_2;\quad \not\subset P_1,\ P_2$ $\Rightarrow (3) \text{ cannot be an edge } \quad ✗$ $(4)\ \vec{d} = (1,-2,3):\quad \vec{n}_2 \cdot \vec{d} = -2-10+12 = 0$ $\Rightarrow \text{line} \subset P_2,\ \text{meets } L \Rightarrow \text{edge of the face on } P_2 \quad ✓$ $\therefore (1),\ (2),\ (4) \text{ are correct}$