Question

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

Answer 1

The lines `(x - x_1)/l_1 = (y - y_1)/m_1 = (z - z_1)/n_1 and (x - x_2)/(l_2) = (y - y_2)/m_2 = (z - z_2)/n_2`

intersect, if `|(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

The equations of the given lines are

`(x - 1)/(2) = (y +1)/3 = (z - 1)/4 and (x - 3)/1 = (y - k)/2 = z/1`

∴ x₁ = 1, y₁ = –1, z₁ = 1, x₂ = 3, y₂ = k, z₂ = 0,

l₁ = 2, m₁ = 3, n₁ = 4, l₂ = 1, m₂ = 2, n₂ = 1.

Since these lines intersect, we get

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

∴ 2(3 – 8) – (k + 1)(2 – 4) – 1(4 – 3) = 0

∴ – 10 + 2(k + 1) – 1 = 0

∴ 2(k + 1) = 11

∴ k + 1  `11/2`

∴ k = `9/2`