Question

If lines `(x - 1)/2 = ("y" + 1)/3 = ("z" - 1)/4` and `(x - 3)/1 = ("y" - lambda)/2 = "z"/1` intersect each other, then λ = ______.

विकल्प

• `7/2`

• `3/2`

• `9/2`

• `5/2`

Answer 1

If lines `(x - 1)/2 = ("y" + 1)/3 = ("z" - 1)/4` and `(x - 3)/1 = ("y" - lambda)/2 = "z"/1` intersect each other, then λ = `underline(9/2)`.

Explanation:

Given lines are

`(x - 1)/2 = ("y" + 1)/3 = ("z" - 1)/4 = lambda_1` (let)     ...(i)

and `(x - 3)/1 = ("y" - lambda)/2 = "z"/1 = lambda_2` (let)     ...(ii)

Then, any point on line (i) is

(2λ₁ + 1, 3λ₁ - 1, 4λ₁ + 1) and any point on line (ii) is (λ₂ + 3, 2λ₂ + λ, λ₂)

Clearly, then lines (i) and (ii) will intersect If

(2λ₁ + 1, 3λ₁ - 1, 4λ₁ + 1) = (λ₂ + 3, 2λ₂ + λ, λ₂)

For some particular value of λ₁ and λ₂ 

⇒ 2λ₁ + 1 = λ₂ + 3, 3λ₁ - 1 = 2λ₂ + λ and 4λ₁ + 1 = λ₂

⇒ 2λ₁ - λ₂ = 2, 3λ₁ - 2λ₂ = λ + 1 and 4λ₁ - λ₂ = - 1

On solving 2λ₁ - λ₂ = 2 and 4λ₁ - λ₂ = - 1

We get `lambda_1 = - 3/2` and λ₂ = - 5

Now. putting the values of λ₁ and λ₂ in

3λ₁ - 2λ₂ = λ + 1 

⇒ `3((-3)/2) - 2(-5) = lambda + 1 => (- 9)/2 + 10 = lambda + 1`

⇒ `lambda + 1 = 11/2`

⇒ `lambda = 9/2`