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JEE Advanced2026MathematicsThree Dimensional GeometryActual

Let L be the straight line joining the points P(1, 2, -1) and Q(2, 3, 1) . Let S be the foot of the perpendicular drawn from the point R(4, -1, 5) to the line L . Another line passing through R intersects L at a point T such that the point S divides the line segment PT internally in the ratio |PS| : |ST| = 1 : 2 , where |PS| and |ST| are the lengths of the line segments PS and ST , respectively. Then which of the fol

Options

  1. AThe orthocentre of the triangle PRT is ( 23 5 , -4, 31 5 )
  2. BThe orthocentre of the triangle PRT is (4, 3, 5)
  3. CThe area of the triangle PRT is 6 5
  4. DThe area of the triangle PRT is 18 5

Correct answer

A. The orthocentre of the triangle PRT is ( 23 5 , -4, 31 5 )

Step-by-step solution

The direction ratios of line L passing through P(1, 2, -1) and Q(2, 3, 1) are (2-1, 3-2, 1-(-1)) = (1, 1, 2) . The equation of line L is x-1 1 = y-2 1 = z+1 2 = . Any point on L can be written as S( +1, +2, 2 -1) . Since S is the foot of the perpendicular from R(4, -1, 5) to L , the direction ratios of RS are ( -3, +3, 2 -6) . As RS L , the dot product of their direction ratios is zero: 1( -3) + 1( +3) + 2(2 -6) = 0 6 - 12 = 0 = 2 Thus, the coordinates of S are (3, 4, 3) . The length of PS is (3-1)^2 + (4-2)^2 + (3

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