Let 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…

Question

Let 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 R^3. Let S = \ X ∈ R^3 : (dist(X,P) )^2 - (dist(X,Q) )^2 = 50 \ and T = \ Y ∈ R^3 : (dist(Y,Q) )^2 - (dist(Y,P) )^2 = 50 \. Then which of the following statements is (are) TRUE?

Accepted Answer

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

Accepted Answer

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

Accepted Answer

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

Answer

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

JEE Advanced 2024 Mathematics question on Three Dimensional Geometry