Question

Consider a plane 2x + y – 3z = 5 and the point P(–1, 3, 2). A line L has the equation `(x - 2)/3 = (y - 1)/2 = (z - 3)/4`. The co-ordinates of a point Q of the line L such that `vec(PQ)` is parallel to the given plane are (α, β, γ), then the product βγ is ______.

Options

• 6.00

• 7.00

• 8.00

• 9.00

Answer 1

Consider a plane 2x + y – 3z = 5 and the point P(–1, 3, 2). A line L has the equation `(x - 2)/3 = (y - 1)/2 = (z - 3)/4`. The co-ordinates of a point Q of the line L such that `vec(PQ)` is parallel to the given plane are (α, β, γ), then the product βγ is 6.00.

Explanation:

Let Q ≡ (3λ + 2, 2λ + 1, 4λ + 3)

`vec(PQ)` = (3λ + 3, 2λ − 2, 4λ + 1)

Since `vec(PQ)` is parallel to plane, 2x + y – 3z = 5

Thus, `vec(PQ)` is perpendicular to the normal of the plane.

⇒ 2(3λ + 3) + (2λ – 2) – 3(4λ + 1) = 0

⇒ λ = `1/4`

∴ Q ≡ `(11/4, 6/4, 4)`

∴ βγ = 6