Vectors & 3D Geometry
Sets S and T as planes — geometric properties
MJAT_TS7_P1
Grade 12
Question:
Let $P=(1,2,3)$, $Q=(4,2,7)$. Define $S=\{X\in\mathbb{R}^3: d(X,P)^2-d(X,Q)^2=50\}$ and $T=\{Y\in\mathbb{R}^3: d(Y,Q)^2-d(Y,P)^2=50\}$. Which is/are TRUE?
A) There is a triangle of area 1 with all vertices from $S$
B) There are distinct $L,M\in T$ such that every point on segment $LM$ is in $T$
C) There are infinitely many rectangles of perimeter 48 with two vertices from $S$ and two from $T$
D) There is a square of perimeter 48 with two vertices from $S$ and two from $T$
Step-by-Step Solution
Key Concept: $S$ and $T$ are planes (loci of points with equal difference of squared distances are planes perpendicular to $PQ$). $S: 6x-8z+d_1=0$ (simplified) and $T$ is the parallel plane. Both are planes.
All four: **ABCD**.
Correct Answer: ABCD