If two straight lines whose direction cosines are given by the relations l + m - n = 0, 3l² + m² + cnl = 0 are parallel, then the positive value of c is :

Solution

Maths sir · 4 steps

Parallel lines share one direction, so the quadratic in l/m needs equal roots: discriminant zero.

1. Remove n. The linear relation gives n = l + m. Put this into the second relation: 3l² + m² + cl(l + m) = 0.

   (3 + c) l² + c lm + m² = 0

2. Equal roots. Read this as a quadratic in l/m. Its two roots give the directions of the two lines. Parallel lines have the same direction, so the roots must be equal and the discriminant must be zero.
3. Solve for c. c² - 4(3 + c) = 0 gives c² - 4c - 12 = 0, that is (c - 6)(c + 2) = 0. So c = 6 or c = -2.
4. So. The positive value is c = 6. As a check, the equation becomes 9l² + 6lm + m² = (3l + m)² = 0, which gives the single direction 1 : -3 : -2. The answer is (a).

The idea to keep. When one linear and one quadratic relation fix the direction cosines, eliminate one variable. Parallel lines mean equal roots, so discriminant zero. Perpendicular lines mean l₁l₂ + m₁m₂ + n₁n₂ = 0.

Answer from the NTA final answer key. Solution by PrepWiser.