Question

If the lines (x - 2)/1 = (y - 3)/1 = (z - 4)/-k and (x - 1)/k = (y - 4)/2 = (z - 5)/1 are coplanar, then find the value of k.

Answer 1

From the lines we have, 

(x₁, y₁, z₁) = (2, 3, 4) and (b₁, b₂, b₃) = (1, 1, -1) 

(x₂, y₂, z₂) = (1, 4, 5) and (d₁, d₂, d₃) = (k, 2, 1) 

Condition for coplanarity

-1(1 + 2k) – 1(1 + k²) + 1(2 – k) = 0 

-1 – 2k – 1 – k² + 2 – k = 0 

-k² – 3k = 0 

k² + 3k = 0 

k(k + 3) = 0 

k = 0 or k = -3