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In three-dimensional space R³, consider the points P = (1, 2, 3) and Q = (4, 2, 7). The distance between two points X and Y is denoted by dist(X, Y). Define the set S as {X ∈ R³: (dist(X, P))² − (dist(X, Q))² = 50} and the set T as {Y ∈ R³: (dist(Y, Q))² − (dist(Y, P))² = 50}. Which of the following statements is/are correct?

1. A triangle with an area of 1 can be formed using only points from S as its vertices.
2. There exist two distinct points L and M in T such that every point on the line segment connecting L and M also belongs to T.
3. It is possible to construct infinitely many rectangles with a perimeter of 48, where two vertices come from S and the other two vertices come from T.
4. A square with a perimeter of 48 can be formed, where two of its vertices are from S and the other two are from T.

Correct answer: A triangle with an area of 1 can be formed using only points from S as its vertices.

Solution

A triangle with an area of 1 can be formed using only points from S as its vertices because the set S is defined by the equation (dist(X, P))² − (dist(X, Q))² = 50, which represents a hyperboloid, and it is possible to find three points on this hyperboloid that form a triangle with an area of 1.

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