Question

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

विकल्प

• `21/2`

• `59/8`

• `57/8`

• `95/8`

Answer 1

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

Explanation:

Given points are

Α(2, 3,9); Β(5, 2, 1); C(1, λ, 8); D(λ, 2, 3)

`[bar(AB)  bar(AC)  bar(AD)] = 0`

`[(3, -1, -8),(-1, lambda-3, -1),(lambda-2, -1, -6)] = 0`

⇒ [−6(λ − 3)−1] −8(1−(λ − 3)(−2)) + (6 + (−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`