Question

If the angle between the line x = `("y" - 1)/2 = ("z - 3")/lambda` and the plane x + 2y + 3z = 4 is `cos^-1 (sqrt(5/14))`, then λ equals.

Options

• `2/3`

• `3/2

• `2/5`

• `5/3`

Answer 1

`2/3`

Explanation:

The d.r.s. of line are 1, 2, λ and

The d.r.s. of normal to the plane are 1, 2, 3.

`therefore sin theta = |(1(1) + 2(2) + lambda(3))/(sqrt(1 + 4 + 9)sqrt(1 + 4 + lambda^2))|`

`=> sin theta = |(5 + 3lambda)/(sqrt14 sqrt(5 + lambda^2))|`

`=> sin^2 theta = (5 + 3lambda)^2/(14 (5 + lambda^2))`

`=> 1 - 5/14 = (5 + 3lambda)^2/(14 (5 + lambda^2))   ....[because cos theta = sqrt(5/14) ("given")]`

`= 9/14 = (25 + 30lambda + 9lambda^2)/(14(5 + lambda^2))`

On solving, we get

`lambda = 2/3`