3D Geometry

If the angle between the line x = (y - 1)/2 = (z - 3)/λ and the plane x + 2y + 3z = 4 is cos-1(5/14) then λ is equal to

  1. 3/2
  2. 2/5
  3. 2/3
  4. 5/3

Answer

(x - 0)/1 = (y - 1)/2 = (z - 3)/λ

x + 2y + 3z = 4

Angle between the line and plane is

cos(90 - θ) = (a1a2 + b1b2 + c1c2) / √(a12 + b12 + c12)√(a22 + b22 + c22)

sinθ = 1 + 4 +3λ / √14 x √(5 + λ2) = 5 + 3λ / √14 x √(5 + λ2)

But given that angle between line and plane is

 θ = cos-1(5/√14) = sin-1(3/√14)

sinθ = 3/√14

3/√14 = 5 + 3λ / √14 x √(5 + λ2))

The correct option is C.