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The lines \( \Large ax+by+c=0,\ bx+cy+a=0\ and\ cx+ay+b=0 \) \( \Large a \ne b \ne c \) are concurrent if:


A) \( \Large a^{3}+b^{3}+c^{3}+3abc=0 \)

B) \( \Large a^{2}+b^{2}+c^{2}-3abc=0 \)

C) \( \Large a+b+c=0 \)

D) none of these

Correct Answer:
C) \( \Large a+b+c=0 \)

Description for Correct answer:

Since, the given lines are concurrent
\begin{vmatrix} 
a & b & c \\ 
b & c & a \\ 
c & a & b  
\end{vmatrix}  = 0 =>\( \Large a^{3}+b^{3}+c^{3}-3abc=0 \)

\( \Large =>  \left(a + b + c\right)  \left(a^{2} + b^{2} + c^{2} - ab - bc - ca\right) = 0   \)

\( \Large =>  \frac{ \left(a + b + c\right) }{2}\{ \left(a-b\right)^{2} +  \left(b - c\right)^{2} +  \left(c - a\right)^{2} = 0    \} \)

\( \Large => a + b + c = 0 \)


Part of solved Straight lines questions and answers : >> Elementary Mathematics >> Straight lines




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