Question

The line $ax+by+c=0,$ $bx+cy+a=0$ and $cx+ay+b=0$ are concurrent if

• A. $\sum _{}^{}{a}^3=3abc$
• B. $\sum _{}^{}{a}^2=abc$
• C. $\sum _{}^{}{a}^2=\sum _{}^{}ab$
• D. none of these

Answer 1

The correct option is A $\sum _{}^{}{a}^3=3abc$

Representing coefficient of three lines in a determinant,

condition for three lines to be concurrent is

$\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|=0$

Solving the determinant,

$=>a(bc-a^2)+b(ac-b^2)+c(ab-c^2)=0$

$=>a^3+b^3+c^3=3abc$