Let 3 denote the three-dimensional space. Take two points P = (1, 2, 3) and Q = (4, 2…

Mathematics · JEE Advanced · NTA Exams — Co-ordinate Geometry

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

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

Answer

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

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