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JEE MainMathematicsThree Dimensional Geometry

The lines L₁: x-1 1 = y-2 -1 = z-3 2 and L₂: x-1 2 = y-a -1 = z-b 1 intersect at a point P . If the perpendicular distance of P from the line L₃: x-2 2 = y-1 1 = z -1 is 5 3 , and S is the sum of all possible values of (a+b) , then the value of 7S is equal to _____.

Correct answer

62

Step-by-step solution

Any point on L₁ can be represented as P( + 1, - + 2, 2 + 3) . The line L₃ passes through the point A(2, 1, 0) and is parallel to the vector v = 2 i + j - k . The vector AP is given by: AP = ( + 1 - 2) i + (- + 2 - 1) j + (2 + 3 - 0) k = ( - 1) i + (- + 1) j + (2 + 3) k The perpendicular distance l from P to L₃ is l = | AP v | | v | . AP v = vmatrix i & j & k - 1 & - + 1 & 2 + 3 2 & 1 & -1 vmatrix = i ( - 1 - 2 - 3) - j (- + 1 - 4 - 6) + k ( - 1 + 2 - 2) = (- - 4) i + (5 + 5) j + (3 - 3) k | AP v |^2 = (- - 4)^2 + (

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