Question

If the lines $ax+by+c=0,bx+cy+a=0$ and $cx+ay+b=0$ are concurrent, then:

• A. $a^3+b^3+c^3+3abc=0$
• B. $a^3+b^3+c^3-abc=0$
• C. $a^3+b^3+c^3-3abc=0$
• D. None of these

Answer 1

The correct option is C

$a^3+b^3+c^3-3abc=0$

Explanation for the correct option:

Concurrent lines:

Given lines $ax+by+c=0,bx+cy+a=0$ and $cx+ay+b=0$ are concurrent.

So, the determinant will be zero.

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

Upon solving we get,

$$\begin{gathered} \Rightarrow a(bc-a^2)-b(b^2-ac)+c(ab-c^2)=0 \\ \Rightarrow abc-a^3-b^3+abc+abc-c^3=0 \\ \Rightarrow a^3+b^3+c^3-3abc=0 \end{gathered}$$

Therefore, the correct answer is option (C).