Question

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=\left\\{X\in\R^3:(dist(X,P)^2)-(dist(X,Q))^2=50\right\\}\ and
T=\left\\{Y\in \R^3:(dist(Y,Q))^2-(dist(Y,P))^2=50\right\\}
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 LM 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.

Answer 1

Answer: (A), (B), (C)

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

Answer 2

The correct option is (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 LM is also in T. and (C): There are infinitely many rectangles of perimeter 48, two of whose vertices are from S and the other two vertices are from T.