If the angle between the line x = (y - 1)/2 = (z - 3)/λ and the plane

• 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.