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?

JEE Advanced 2024 Mathematics question

Let denote the three-dimensional space. Take two points and . Let denote the distance between two points and in . Let

S = {X : (dist(X, - (dist(X, = 50} and

T = {Y : (dist(Y, - (dist(Y, = 50}.

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 .Correct
(B)There are two distinct points and in such that each point on the line segment is also in .Correct
(C)There are infinitely many rectangles of perimeter 48, two of whose vertices are from and the other two vertices are from .Correct
(D)There is a square of perimeter 48, two of whose vertices are from and the other two vertices are from .Correct

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

Solution

with , , . So S is the plane , i.e. , and T is the plane . They are parallel planes at distance . (A) A plane contains triangles of every area – TRUE. (B) T is a plane,

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Question ID VK-034170