Question

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

• A. $\sum a^3=3abc$
• B. $\sum a^2=abc$
• C. $\sum a^3=\sum bc$
• D. None of these

Answer 1

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

Given the equations of the lines are,

$ax+by+c=0$......(1)

$bx+cy+a=0$......(2)

$cx+ay+b=0$......(3).

Now these three lines will be concurrent if,

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

Or, $(abc-a^3)-(b^3-abc)+(abc-c^3)=0$

Or, $a^3+b^3+c^3=3abc$

Or, $\sum a^3=3abc$.