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JEE Main202311 Apr 2023Evening ShiftMathematicsThree Dimensional GeometryActual

Let the line L : x = 1 - y - 2 = z - 3 λ , λ ∈ ℝ meet the plane P : x + 2 y + 3 z = 4 at the point ( α , β , γ ) . If the angle between the line L and the plane P is cos - 1 5 14 , then α + 2 β + 6 γ is equal to

Correct answer

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Step-by-step solution

The given equations are, L : x - 0 1 = y - 1 2 = z - 3 λ   &     P : x + 2 y + 3 z = 4 So, Vector parallel to line : < 1 ,   2 ,   λ > = b → Normal vector to plane P : < 1 ,   2 ,   3 > = n → Angle between plane & line is θ . Then angle between the line and normal to the plane will be 90 - θ . Then, ⇒ cos 90 - θ = i ^ + 2 j ^ + λ k ^ · i ^ + 2 j ^ + 3 k ^ 1 2 + 2 2 + λ 2 · 1 2 + 2 2 + 3 2 ⇒ 3

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