Let \mathbb{R}^3 denote the three-dimensional space. Take two points P=(1,2,3) and Q=(4,2,7). Let {\dist}(X, Y) denote the distance between two points X and Y in \mathbb{R}^3. Let S= \{X \in \mathbb{R}^3:({\dist}(X, P))^2-({\dist}(X, Q))^2=50\} and T= \{Y \in \mathbb{R}^3:({\dist}(Y, Q))^2-({\dist}(Y, P))^2=50\}. Then which of the following statements is (are) TRUE? [JEE Adv 2024 P1]

There is a triangle whose area is 1 and all of whose vertices are from S.

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

There are infinitely many rectangles of perimeter 48 , two of whose vertices are from S and the other two vertices are from T.

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

Three Dimensional Geometry

Section: Mathematics

Question : 3 of 35

Marks: +1, -0

Let ℝ3 denote the three-dimensional space. Take two points P=(1,2,3) and Q=(4,2,7). Let dist(X,Y) denote the distance between two points X and Y in ℝ3. Let
S={X∈ℝ3:(dist(X,P))2−(dist(X,Q))2=50} and
T={Y∈ℝ3:(dist(Y,Q))2−(dist(Y,P))2=50}.
Then which of the following statements is (are) TRUE?

[JEE Adv 2024 P1]

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

Validate

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