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JEE MainMathematicsThree Dimensional Geometry

Let a point P lie on the line L₃: x-1 1 = y+1 2 = z 1 . Let Q be the foot of the perpendicular drawn from P to the plane : 2x - z = 0 , and R be the foot of the perpendicular drawn from P to the line L₁: x-1 1 = y-2 1 = z-3 -1 . If QPR is a right angle, then the square of the distance of P from the origin is ________

Correct answer

50

Step-by-step solution

Let the coordinates of P on the line L₃ be ( + 1, 2 - 1, ) . Since Q is the foot of the perpendicular from P to the plane : 2x - z = 0 , the vector PQ is parallel to the normal vector of the plane, n = 2 i - k . Let the coordinates of R on the line L₁ be ( + 1, + 2, - + 3) . The vector PR = ( - ) i + ( - 2 + 3) j + (- - + 3) k . Since R is the foot of the perpendicular from P to L₁ , PR is perpendicular to the direction of L₁ (which is i + j - k ): PR ( i + j - k ) = 0 1( - ) + 1( - 2 + 3) - 1(- - + 3) = 0 - + - 2

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