Line through the intersection of two lines — A/L Combined Mathematics

What must I remember about Line through the intersection of two lines?

• Intersection of two lines: If L1: a1*x + b1*y + c1 = 0 and L2: a2*x + b2*y + c2 = 0 intersect at point P(x0, y0), the coordinates satisfy both equations simultaneously.
• Equation of line bundle (pencil of lines): Any line passing through the point of intersection of L1 = 0 and L2 = 0 can be written as L1 + λ*L2 = 0, where λ is a real parameter (excluding L2 itself).
• Expanded pencil equation: (a1 + λ*a2)x + (b1 + λ*b2)y + (c1 + λ*c2) = 0; represents a straight line for every real scalar λ.
• Determining parameter λ: Parameter λ is determined using an auxiliary geometrical condition (e.g., passing through a given external point, having a specified gradient, being parallel or perpendicular to a given line).
• Concurrent lines: Three lines L1 = 0, L2 = 0, L3 = 0 are concurrent (intersect at a single common point) if and only if there exist constants l, m, n (not all zero) such that l*L1 + m*L2 + n*L3 ≡ 0, or determinant of coefficients det([a, b, c]) = 0.

Test yourself (1 minute)

1. Avoid computing intersection coordinates

   Using L1 + λ*L2 = 0 avoids the clumsy, tedious step of finding explicit coordinates (x0, y0), drastically reducing algebra and preventing arithmetic errors.

2. Handling missing L2 line

   If the required line is L2 itself, λ diverges to infinity; use symmetric formulation l*L1 + m*L2 = 0 where both lines are naturally included.

3. Concurrency determinant condition

   Lines a1*x + b1*y + c1 = 0, a2*x + b2*y + c2 = 0, a3*x + b3*y + c3 = 0 are concurrent iff: |a1 b1 c1; a2 b2 c2; a3 b3 c3| = 0.

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Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 18

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Year 1 · Term 3