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JEE MainMathematicsThree Dimensional Geometry

Let a be a positive integer. If the lines r = (a i + j ) + (2 i + j + k ) and r = ( i + a j ) + ( i + 2 j + k ) intersect, and the square of the distance of their point of intersection from the origin is 17 , then the value of a is:

Options

  1. A4
  2. B10
  3. C2
  4. D10 3

Correct answer

A. 4

Step-by-step solution

Equating the position vectors of the two lines at their point of intersection: (a + 2 ) i + (1 + ) j + k = (1 + ) i + (a + 2 ) j + k Comparing the coefficients of i , j , k : a + 2 = 1 + 1 + = a + 2 = Substituting = into the first equation: a + 2 = 1 + = 1 - a The coordinates of the point of intersection are: x = a + 2(1 - a) = 2 - a y = 1 + (1 - a) = 2 - a z = 1 - a The square of the distance from the origin is given as 17 : (2 - a)^2 + (2 - a)^2 + (1 - a)^2 = 17 2(4 - 4a + a^2) + (1 - 2a + a^2) = 17 8 - 8a + 2a^2

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