Question

If the lines `(x - 1)/(2) = (y + 1)/(3) = (z -1)/(4) and (x- 2)/(1) = (y +m)/(2) = (z - 2)/(1)` intersect each other, find m.

Answer 1

The lines `(x - 1)/(2) = (y + 1)/(3) = (z -1)/(4) and (x- 2)/(1) = (y +m)/(2) = (z - 2)/(1)` intersect, if 

∴ `|(x_2 - x_1,y_2 - y_2,z - z_2),(a_1, b_1, c_1),(a_2, b_2, c_2)|` = 0                 ...(1)

Here, (x₁, y₁, z₁) ≡ (1, – 1, 1),
(x₂, y₂, z₂) ≡ (2, – m, 2),
a₁ = 2, b₁ = 3, c₁ = 4,
a₂ = 1, b₂ = 2, c₂ = 1
Substituting these values in (1), we get

`|(2 - 1, -m + 1, 2 - 1),(2, 3, 4),(1, 2, 1)|` = 0

∴ `|(1, 1 - m, 1),(2, 3, 4),(1, 2, 1)|` = 0

∴ 1(3 – 8) – (1 – m)(2 – 4) + 1(4 – 3) = 0
∴ –5 + 2 – 2m + 1 = 0
∴ – 2m = 2
∴ m = – 1.