Question

Let (λ, 2, 1) be a point on the plane which passes through the point (4, –2, 2). If the plane is perpendicular to the line joining the points (–2, –21, 29) and (–1, –16, 23), then `(λ/11)^2 - (4λ)/11 - 4` is equal to ______.

Options

• 6

• 7

• 8

• 9

Answer 1

Let (λ, 2, 1) be a point on the plane which passes through the point (4, –2, 2). If the plane is perpendicular to the line joining the points (–2, –21, 29) and (–1, –16, 23), then `(λ/11)^2 - (4λ)/11 - 4` is equal to 8.

Explanation:

Since the plane is perpendicular to the line joining points

(–2, –21, 29) and (–1, –16, 23)

∴ Normal vector of plane is

`overlinen = hati + 5hatj - 6hatk`

Let A(λ, 2, 1)  and (4, –2, 2)

∵ `overline(AB) ⊥ overlinen`

`\implies` (λ – 4) + 5 × 4 – 6(–1) = 0

`\implies` λ – 4 + 20 + 6 = 0

`\implies` λ = – 22

Hence, `(λ/11)^2 - 4(λ/11) - 4` = 8