Question

If the straight lines `(x - 1)/2 = (y + 1)/lambda = z/2` and `(x + 1)/5 = (y + 1)/2 = z/lambda` are coplanar, find λ and equations of the planes containing these two lines

Answer 1

If the two lines are coplanar

`|(x_2 - x_1, y_2 - y_1, z_2 - z_1),(l_1, "m"_1, "n"_1),("l"_2, "m"_2, "n"_2)|` = 0

(x₁, y₁, z₁) = (1, −1, 0), (x₂, y₂, z₂) = (−1, −1, 0)

(l₁, m₁, n₁) = (2, λ, 2), (l₂, m₂, n₂) = (5, 2, λ)

⇒ `|(-2, 0, 0),(2, lambda, 2),(5, 2, lambda)|` = 0

When λ = 2

(x₁, y₁, z₁) = (1, −1, 0)

(b₁, b₂, b₃) = (2, 2, 2)

(d₁, d₂, d₃) = (5, 2, 2)

`|(x - x_1, y - y_1, z - z_1),("b"_1, "b"_2, "b"_3),("d"_1, "d"_2, "d"_3)|` = 0

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

⇒ (x – 1)(0) – (y + 1)(– 6) + z(6) = 0

⇒ 6(y + 1) – 6z = 0

⇒ 6y + 6 – 6z = 0

⇒ y – z + 1 = 0

When λ = 2

(b₁, b₂, b₃) = (2, – 2, 2)

(d₁, d₂, d₃) = (5, 2, – 2)

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

⇒ (x – 1)(0) – (y + 1)(– 14) + z(4 + 10) = 0

⇒ 14(y + 1) + 14z = 0

⇒ 14y + 14 + 14z = 0

⇒ y + z + 1 = 0