Question

If (2, 3, 9), (5, 2, 1), (1, λ, 8) and (λ, 2, 3) are coplanar, then the product of all possible values of λ is ______.

Options

• `21/2`

• `59/8`

• `57/8`

• `95/8`

Answer 1

If (2, 3, 9), (5, 2, 1), (1, λ, 8) and (λ, 2, 3) are coplanar, then the product of all possible values of λ is `underlinebb(95/8)`.

Explanation:

Given points are A(2, 3, 9); B(5, 2, 1); C(1, λ, 8); D(λ, 2, 3)

`[(vec(AB), vec(AC), vec(AD))]` = 0

`|(3, -1, -8),(-1, λ - 3, -1),(λ - 2, -1, -6)|` = 0

`\implies ` [–6(λ – 3)–1] –8(1–(λ – 3)(λ – 2)) + (6 + (λ – 2) = 0

3(–6λ + 17) – 8(–λ² + 5λ – 5) + (λ + 4) = 8

8λ² – 57λ + 95 = 0

Apply the rule of product whose roots are αβ

αβ = `95/8`