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3D Geometry question

2024 · Shift 1 · Q24

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PYQ Library
JEE Advanced
Mathematics
3D Geometry
2024 · Shift 1 · Q24

3D Geometry question

2024 · Shift 1 · Q24

JEE AdvancedMathematics3D GeometryMultiple correct+4 / −2

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?

A
There is a triangle whose area is 1 and all of whose vertices are from $S$ .
B
There are two distinct points $L$ and $M$ in $T$ such that each point on the line segment $L M$ is also in $T$ .
C
There are infinitely many rectangles of perimeter 48 , two of whose vertices are from $S$ and the other two vertices are from $T$ .
D
There is a square of perimeter 48 , two of whose vertices are from $S$ and the other two vertices are from $T$ .

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