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JEE Main202227 Jul 2022Morning ShiftMathematicsThree Dimensional GeometryActual

Let the line x - 3 7 = y - 2 - 1 = z - 3 - 4 intersect the plane containing the lines x - 4 1 = y + 1 - 2 = z 1 and 4 a x - y + 5 z - 7 a = 0 = 2 x - 5 y - z - 3 , a ∈ ℝ at the point P α , β , γ . Then the value of α + β + γ equals ______.

Correct answer

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Step-by-step solution

Given plane contain line x - 4 1 = y + 1 - 2 = z 1 and 4 a x - y + 5 z - 7 a = 0 = 2 x - 5 y - z - 3 So equation of plane A 4 a x - y + 5 z - 7 a + λ 2 x - 5 y - z - 3 = 0 will satisfy 4 , - 1 , 0 ⇒ 16 a + 1 - 7 a + λ 8 + 5 - 3 = 0 ⇒ 9 a + 1 + 10 λ = 0             ⋯ 1 Now normal vector of the plane A is 4 a + 2 λ , - 1 - 5 λ , 5 - λ and direction ratio along the line which contained the plane A is i - 2 j + k Since they will be perpendicu

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