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JEE Main202427 Jan 2024Evening ShiftMathematicsThree Dimensional GeometryActual

The lines x - 2 2 = y - 2 = z - 7 16 and x + 3 4 = y + 2 3 = z + 2 1 intersect at the point P . If the distance of P from the line x + 1 2 = y - 1 3 = z - 1 1 is l , then 14 l 2 is equal to _____.

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Step-by-step solution

Given: x - 2 2 = y - 2 = z - 7 16 and x + 3 4 = y + 2 3 = z + 2 1 ⇒ x - 2 1 = y - 1 = z - 7 8 = λ let and x + 3 4 = y + 2 3 = z + 2 1 = k let ⇒ x = λ + 2 = 4 k - 3 , y = - λ = 3 k - 2 and z = 8 λ + 7 = k - 2 ⇒ λ = - 5 + 4 k , λ = - 3 k + 2 , 8 λ - k = - 9 ⇒ - 5 + 4 k = - 3 k + 2 ⇒ 7 k = 7 ⇒ k = 1 , λ = - 3 1 + 2 = - 1 ⇒ x = 1 , y = 1 , z = - 1 ⇒ P ≡ (1,1,-1) Projection of PQ ≡ 2 i ^ - 2 k ^ on QR line with direction ratio 2 i ^ + 3 j ^ + k ^ is given by, QR = 4 - 2 4 + 9 + 1 = 2 14 Now, in triangle PQR we get, ∴ l

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