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JEE MainMathematicsThree Dimensional Geometry

Let the point A be (4, 3, 4) . The points B and C lie on the line x 1 = y 2 = z 2 . If B is the foot of the perpendicular drawn from A to this line and the area of ABC is 3 5 sq. units, then the maximum possible distance of point C from the origin is :

Options

  1. A9
  2. B12
  3. C20
  4. D24

Correct answer

B. 12

Step-by-step solution

Let the coordinates of B be ( , 2 , 2 ) since it lies on the given line. The direction ratios of the line are 1, 2, 2 . The vector AB is ( - 4) i + (2 - 3) j + (2 - 4) k . Since B is the foot of the perpendicular from A to the line, AB is perpendicular to the line's direction vector. Therefore, their dot product is zero: 1( - 4) + 2(2 - 3) + 2(2 - 4) = 0 - 4 + 4 - 6 + 4 - 8 = 0 9 - 18 = 0 = 2 Thus, the coordinates of B are (2, 4, 4) . The length of the perpendicular AB is: AB = (2 - 4)^2 + (4 - 3)^2 + (4 - 4)^2 = 4

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