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JEE Main202429 Jan 2024Evening ShiftMathematicsThree Dimensional GeometryActual

Let O be the origin, and M and N be the points on the lines x - 5 4 = y - 4 1 = z - 5 3 and x + 8 12 = y + 2 5 = z + 11 9 respectively such that M N is the shortest distance between the given lines. Then O M → · O N → is equal to _________.

Correct answer

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Step-by-step solution

Let, L 1 : x - 5 4 = y - 4 1 = z - 5 3 = λ ⇒ M ( 4 λ + 5 , λ + 4 , 3 λ + 5 ) And L 2 : x + 8 12 = y + 2 5 = z + 11 9 = μ ⇒ N ( 12 μ - 8 , 5 μ - 2 , 9 μ - 11 ) Now, finding M N → = ( 4 λ - 12 μ + 13 , λ - 5 μ + 6 , 3 λ - 9 μ + 16 ) . . … 1 Now finding cross product of direction ratios of line we get, b → 1 × b → 2 = i ^ j ^ k ^ 4 1 3 12 5 9 = - 6 i ^ + 8 k ^ … . . . 2 Now, solving the equation 1 & 2 as they are parallel vectors, we get, ∴ 4 λ - 12 μ + 13 - 6 = λ - 5 μ + 6 0 = 3 λ - 9 μ + 16 8 Taking 4 λ - 12 μ + 13

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