If u = a₁x + b₁y + c₁ = 0, v = a₂x + b₂y + c₂ = 0 and (a₁ / a₂) = (b₁ / b₂) =…
2019
If u = a₁x + b₁y + c₁ = 0, v = a₂x + b₂y + c₂ = 0 and (a₁ / a₂) = (b₁ / b₂) = (c₁ / c₂), then u + kv = 0 represents:
- A.
a family of concurrent lines
- B.
u = 0
- C.
none of these
- D.
a family of parallel lines
Attempted by 65 students.
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Correct answer: B
Key idea: When all three coefficient ratios are equal, a₁/a₂ = b₁/b₂ = c₁/c₂, the two equations u = 0 and v = 0 are coincident — they are the same straight line, not two different lines.
Reasoning:
Let a₁/a₂ = b₁/b₂ = c₁/c₂ = λ. Then a₁ = λa₂, b₁ = λb₂, c₁ = λc₂, so u = a₁x + b₁y + c₁ = λ(a₂x + b₂y + c₂) = λv. In other words v is just a scalar multiple of u.
Substitute into the combination: u + kv = u + k(u/λ) = (1 + k/λ)u.
For every value of k except the single one making the bracket (1 + k/λ) zero, the equation reduces to u = 0 — the same line each time. At that one exceptional k the equation becomes the identity 0 = 0, which adds no new line. So no fresh line is ever produced.
Contrast: u + kv = 0 gives a genuine family only when u = 0 and v = 0 are two DIFFERENT lines that intersect (a family of concurrent lines through their common point). That requires a₁/a₂ ≠ b₁/b₂, which is not the case here.
Conclusion: Because the lines are coincident, for every admissible k the equation u + kv = 0 represents the single line u = 0. The correct choice is u = 0.