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JEE MainMathematicsThree Dimensional Geometry

The lines L₁: x-2 1 = y-1 -1 = z-5 2 and L₂: x-5 2 = y -1 = z-5 1 intersect at a point P . A line L₃ passes through P and the origin (0,0,0) . If d is the shortest distance between the line L₃ and the line L₄: x-2 2 = y-1 1 = z-1 -1 , then the value of 83d^2 is equal to _____.

Correct answer

36

Step-by-step solution

Let the coordinates of any point on L₁ be given by x-2 1 = y-1 -1 = z-5 2 = . So, x = + 2 , y = - + 1 , z = 2 + 5 . Let the coordinates of any point on L₂ be given by x-5 2 = y -1 = z-5 1 = . So, x = 2 + 5 , y = - , z = + 5 . For the point of intersection P , we equate the coordinates: + 2 = 2 + 5 - 2 = 3 - + 1 = - - = 1 Solving these two equations, we subtract the second from the first: - = 2 = -2 Substituting = -2 into - = 1 gives = -1 . Checking the z -coordinates: 2(-1) + 5 = 3 and -2 + 5 = 3 , which match. Thu

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