If the lines and are coplanar, then can have
- A.any value
- B.exactly one value
- C.exactly two values
- D.exactly three values
Show answer and solution
Correct option: C
Solution
For two lines to be coplanar, the scalar triple product of their direction vectors and the vector joining a point on each line must be zero.
Direction vectors: d₁ = (1, 1, k), d₂ = (k, 2, 1). Vector joining points (2,3,4) and (1,4,5): v = (-1, 1, 1).
Condition: v · (d₁ × d₂) = 0.
d₁ × d₂ = |i j k; 1 1 k; k 2 1| = i(1·1 - k·2) - j(1·1 - k·k) + k(1·2 - 1·k) = (1-2k, k²-1, 2-k).
v · (d₁ × d₂) = (-1)(1-2k) + (1)(k²-1) + (1)(2-k) = -1+2k + k²-1 + 2-k = k² + k = k(k+1) = 0.
So k = 0 or k = -1, giving exactly two values.
Direction vectors: d₁ = (1, 1, k), d₂ = (k, 2, 1). Vector joining points (2,3,4) and (1,4,5): v = (-1, 1, 1).
Condition: v · (d₁ × d₂) = 0.
d₁ × d₂ = |i j k; 1 1 k; k 2 1| = i(1·1 - k·2) - j(1·1 - k·k) + k(1·2 - 1·k) = (1-2k, k²-1, 2-k).
v · (d₁ × d₂) = (-1)(1-2k) + (1)(k²-1) + (1)(2-k) = -1+2k + k²-1 + 2-k = k² + k = k(k+1) = 0.
So k = 0 or k = -1, giving exactly two values.
[JEE Main 2012]