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JEE MainMathematicsThree Dimensional Geometry

A line with direction ratios 2, 4, 1 intersects the lines x-2 1 = y-3 2 = z -1 and x-1 2 = y-4 1 = z+1 3 at the points P and Q , respectively. If O is the origin, then the square of the area of the parallelogram whose adjacent sides are represented by the vectors OP and OQ is equal to:

Options

  1. A7
  2. B100
  3. C21
  4. D14

Correct answer

D. 14

Step-by-step solution

Let the coordinates of P on the first line be (2+ , 3+2 , - ) . Let the coordinates of Q on the second line be (1+2 , 4+ , -1+3 ) . The direction ratios of the line segment PQ are proportional to 2, 4, 1 . Thus, we can write: (1+2 ) - (2+ ) 2 = (4+ ) - (3+2 ) 4 = (-1+3 ) - (- ) 1 = k This gives the system of equations: 2 - - 1 = 2k - 2 + 1 = 4k 3 + - 1 = k Substitute k = 3 + - 1 into the first equation: 2 - - 1 = 2(3 + - 1) 2 - - 1 = 6 + 2 - 2 4 + 3 = 1 Substitute k = 3 + - 1 into the second equation: - 2 + 1 = 4(3

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