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JEE Advanced2024MathematicsThree Dimensional GeometryActual

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

Options

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

Correct answer

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

Step-by-step solution

aligned & S = X :( XP )^2-( XQ )^2=50 & T = Y :( YQ )^2-( YP )^2=50 aligned for finding S X(x, y, z) and for T Y(x, y, z) aligned & ((x-1)^2+(y-1)^2+( z -1)^2 )- (( x -4)^2+( y -2)^2+( z -7)^2 )=50 & S = ( x , y , z ): 6 x +8 z =105 & T = ( x , y , z ): 6 x +8 z =5 aligned Since S and T both are plane ; (1) There exist a triangle in plane S whose area =1 (always) (2) L ~ &~ M lies on plane T , hence line segment joining L & M will lie on plane T . (3) Distance between S & ~T d = | 105-5 10 |=10 Hence for rectangle

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