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The three straight lines $a x+b y=c, b x+c y=a$ and $c x+a y=b$ are collinear, if
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The correct answer is:
$a+b+c=0$
Given three straight lines $a x+b y-c=0, b x+c y-a=0, c x+a y-b=0$ are collinear,
$\left|\begin{array}{lll}
a & b & -c \\
b & c & -a \\
c & a & -b
\end{array}\right|=0 \Rightarrow-(a+b+c)\left|\begin{array}{lll}
1 & b & c \\
1 & c & a \\
1 & a & b
\end{array}\right|=0$
Clearly, $(a+b+c)=0$
$\left|\begin{array}{lll}
a & b & -c \\
b & c & -a \\
c & a & -b
\end{array}\right|=0 \Rightarrow-(a+b+c)\left|\begin{array}{lll}
1 & b & c \\
1 & c & a \\
1 & a & b
\end{array}\right|=0$
Clearly, $(a+b+c)=0$
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