Q. The lines $ax + by + c = 0,\ bx + cy + a = 0$ and $cx + ay + b = 0$ $(a \neq b \neq c)$ are concurrent, if
A) $a^{3} + b^{3} + c^{3} + 3abc = 0$
B) $a^{2} + b^{2} + c^{2} - 3abc = 0$
C) $a + b + c = 0$
D) None of the above
Explanation
(c)
Since, the given lines are concurrent
$\therefore\left| \begin{matrix}
a & b & c \\
b & c & a \\
c & a & b
\end{matrix} \right| = 0 \Rightarrow a^{3} + b^{3} + c^{3} - 3abc = 0$
$\Rightarrow (a + b + c)\left( a^{2} + b^{2} + c^{2} - ab - bc - ca \right) = 0$
$\Rightarrow \frac{(a + b + c}{2}\{(a - {b)}^{2} + (b - {c)}^{2} + \left( c - {a)}^{2} \right\} = 0$
$\Rightarrow a + b + c = 0$ $(as\ a \neq b \neq c)$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.