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JEE Main2014MathematicsThree Dimensional GeometryActual

If the angle between the line 2 x ⁡ + 1 = y ⁡ = z ⁡ + 4 and the plane 2 x ⁡ - y ⁡ + λ z ⁡ + 4 = 0 is π 6 , then the value of λ is

Options

  1. A4 5 7
  2. B1 3 5 1 1
  3. C1 3 5 7
  4. D4 5 1 1

Correct answer

A. 4 5 7

Step-by-step solution

Given line is L : x + 1 1 = y 2 = z + 4 2 and plane is P :   2 x - y + λ z + 4 = 0 . The angle between a line x - x 1 l = y - y 1 m = z - z 1 n and a plane a x + b y + c z + d = 0 is sin θ = a l + b m + c n a 2 + b 2 + c 2 l 2 + m 2 + n 2 sin π 6 = 2 × 1 + - 1 × 2 + 2 λ 1 2 + 2 2 + 2 2 2 2 + 1 2 + λ ⇒ 1 2 = 2 - 2 + 2 λ 1 + 4 + 4 4 + 1 + λ ⇒ 1 2 = 2 λ 3 5 + λ ⇒ 4 λ = 3 5 + λ Squaring both sides, we get ⇒ 1 6 λ = 9 5 +

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