Concurrent linesHardQuestion 300011
Question
Lines ax + by + c = 0, bx + cy + a = 0, cx + ay + b = 0 are concurrent if:
- Aa³ + b³ + c³ − 3abc = 0Correct
- Ba + b + c = 0
- Ca = b = c
- Dab + bc + ca = 0
Correct answer
a³ + b³ + c³ − 3abc = 0
Explanation
Three lines are concurrent iff their determinant = 0. Setting up the 3×3 determinant: |a b c; b c a; c a b| = 0 gives a³+b³+c³−3abc = 0 (or equivalently a+b+c = 0 when the factored form (a+b+c)(a²+b²+c²−ab−bc−ca) = 0).