Question

If the straight lines `(x - 1)/2 = (y + 1)/k = z/2` and `(x + 1)/5 = (y + 1)/2 = z/k` are coplanar, then the plane(s) containing these two lines is/are ______.

Options

• y + 2z = – 1

• y + z = – 1

• y – z = – 1

• y – 2z = – 1

Answer 1

If the straight lines `(x - 1)/2 = (y + 1)/k = z/2` and `(x + 1)/5 = (y + 1)/2 = z/k` are coplanar, then the plane(s) containing these two lines is/are y + z = – 1.

Explanation:

If straight lines are coplanar, then

`\implies |(x_2 - x_1, y_2 - y_1, z_2 - z_1),(a_1, b_1, c_1),(a_2, b_2, c_2)|` = 0

Since, `(x - 1)/2 = (y + 1)/k = z/2` and `(x + 1)/5 = (y + 1)/2 = z/k` are coplanar.

`\implies |(2, 0, 0),(2, k, 2),(5, 2, k)|` = 0

`\implies` k² = 4

`\implies` k = ± 2

∴ n₁ = b₁ × d₁ = `6hatj - 6hatk` for k = 2

∴ n₂ = b₂ × d₂ = `14hatj + 14hatk` for k = – 2

So, equation of planes are (r – a) . n₁ = 0

`\implies` y – z = – 1 and (r – a) . n₂ = 0

`\implies` y + z = – 1