Let $\mathbb{R}^3$ denote the three-dimensional space. Take two points $P=(1,2,3)$ and $Q=(4,2,7)$. Let $\operatorname{dist}(X, Y)$ denote the distance between two points $X$ and $Y$ in $\mathbb{R}^3$. Let $\begin{gathered}S=\left\{X \in \mathbb{R}^3:(\operatorname{dist}(X, P))^2-(\operatorname{dist}(X, Q))^2=50\right\} \text { and } \\T=\left\{Y \in \mathbb{R}^3:(\operatorname{dist}(Y, Q))^2-(\operatorname{dist}(Y, P))^2=50\right\} .\end{gathered}$ Then which of the following statements is (are) TRUE?

• Correct answer
  A

  There is a triangle whose area is 1 and all of whose vertices are from $S$.
• Correct answer
  B

  There are two distinct points $L$ and $M$ in $T$ such that each point on the line segment $L M$ is also in $T$.
• Correct answer
  C

  There are infinitely many rectangles of perimeter 48 , two of whose vertices are from $S$ and the other two vertices are from $T$.
• Correct answer
  D

  There is a square of perimeter 48 , two of whose vertices are from $S$ and the other two vertices are from $T$.

Solution

$\begin{aligned}& \mathrm{S}=\left\{\mathrm{X}:(\mathrm{XP})^2-(\mathrm{XQ})^2=50\right\} \\& \mathrm{T}=\left\{\mathrm{Y}:(\mathrm{YQ})^2-(\mathrm{YP})^2=50\right\}\end{aligned}$ for finding $S \equiv X(x, y, z)$ and for $T \equiv Y(x, y, z)$ $\begin{aligned}&\left((x-1)^2+(y-1)^2+(\mathrm{z}-1)^2\right)-\left((\mathrm{x}-4)^2+(\mathrm{y}-2)^2+(\mathrm{z}-7)^2\right)=50 \\& \Rightarrow \quad \mathrm{S}=\{(\mathrm{x}, \mathrm{y}, \mathrm{z}): 6 \mathrm{x}+8 \mathrm{z}=105\} \\& \mathrm{T}=\{(\mathrm{x}, \mathrm{y}, \mathrm{z}): 6 \mathrm{x}+8 \mathrm{z}=5\}\end{aligned}$ Since $\mathrm{S}$ and $\mathrm{T}$ both are plane ; (1) There exist a triangle in plane $\mathrm{S}$ whose area $=1$ (always) (2) $\mathrm{L} ~\&~ \mathrm{M}$ lies on plane $\mathrm{T}$, hence line segment joining $\mathrm{L} \& \mathrm{M}$ will lie on plane $\mathrm{T}$. (3) Distance between $\mathrm{S} \& \mathrm{~T}$ $\mathrm{d}=\left|\frac{105-5}{10}\right|=10$ Hence for rectangle of perimeter 48 can exist. (4) For Square There will be infinite such rectangle possible. Hence Answers 1,2,3,4 are correct.

Hence, the correct answer is options: 'A', 'B', 'C', 'D'.