Question

Let P₁: `vecr.(2hati + hatj - 3hatk)` = 4 be a plane. Let P₂ be another plane which passes through the points (2, –3, 2), (2, –2, –3) and (1, –4, 2). If the direction ratios of the line of intersection of P₁ and P₂ be 16, α, β, then the value of α + β is equal to ______.

Options

• 27

• 28

• 29

• 30

Answer 1

Let P₁: `vecr.(2hati + hatj - 3hatk)` = 4 be a plane. Let P₂ be another plane which passes through the points (2, –3, 2), (2, –2, –3) and (1, –4, 2). If the direction ratios of the line of intersection of P₁ and P₂ be 16, α, β, then the value of α + β is equal to 28.

Explanation:

Given: `P_1: vecr.(2hati + hatj - 3hatk)` = 4

⇒ P₁: 2x + y – 3z = 4

Now, equation of plane passing through points (2, –3, 2), (2, –2, –3) and (1, –4, 2) is given by

`|((x - 2) (y + 3) (z - 2)),((2 - 2) (-2 + 3) (-3 - 2)),((1 - 2) (-4 + 3) (2 - 2))|` = 0

⇒ `|(x - 2, y + 3, z - 2),(0, 1, -5),(-1, -1, 0)|` = 0

⇒ (x – 2)(–5) – (y + 3) (–5) + z – 2) = 0

⇒ –5x + 5y + z + 23 = 0

∴ P₂: – 5x + 5y + z + 23 = 0

Let a, b, c be the direction ratios of the line of intersection of plane P₁ and P₂

∴ `a/(1 + 15) = (-b)/(2 - 15) = c/(10 + 5)` = λ

⇒ a = 16λ, b = 13λ, c = 15λ

⇒ α = 13, β = 15

∴ α + β = 28