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JEE MainMathematicsThree Dimensional Geometry

Let L be the line of intersection of the planes x + y - z = 2 and x - y + z = 4 . If the point A(3, 2, 5) is reflected in the line L to obtain an image point A' , then the perpendicular distance from A' to the plane 2x + 2y - z = 1 is:

Options

  1. A4 3
  2. B10 3
  3. C7 3
  4. D8 3

Correct answer

B. 10 3

Step-by-step solution

First, we find the equation of the line L . The direction vector of L is d = vmatrix i & j & k 1 & 1 & -1 1 & -1 & 1 vmatrix = -2 j - 2 k , which is parallel to (0, 1, 1) . To find a point on L , we can set z = 1 in the plane equations: x + y = 3 x - y = 3 Solving these gives x = 3, y = 0 . So, (3, 0, 1) lies on L . The equation of L is r = 3 i + k + ( j + k ) . Let F(3, , + 1) be the foot of the perpendicular from A(3, 2, 5) to L . The vector AF = (0, - 2, - 4) . Since AF is perpendicular to L , AF (0, 1, 1) = 0 -

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