Straight Lines
Condition of Concurrency
Premium Question
Grade 11
Question:
If the lines $ax + by + c = 0, bx + cy + a = 0$ and $cx + ay + b = 0$ are concurrent, then:
(1) $a + b + c = 0$
(2) $a^3 + b^3 + c^3 - 3abc = 0$
(3) $a^2 + b^2 + c^2 = ab + bc + ca$
(4) Both (1) and (2)
Step-by-Step Solution
Key Concept: Set the $3 \times 3$ determinant of coefficients equal to zero. Expand the determinant and factorise using $a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (4)