A) 0
B) 1
C) \[\frac{1}{a+b+c}\]
D) None of these
Correct Answer: B
Solution :
If the given lines are concurrent, then \[\left| \,\begin{matrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \\ \end{matrix}\, \right|=0\Rightarrow \left| \,\begin{matrix} a & 1-a & 1-a \\ 1 & b-1 & 0 \\ 1 & 0 & c-1 \\ \end{matrix} \right|=0\]{Apply \[{{C}_{2}}\to {{C}_{2}}-{{C}_{1}}\]and\[{{C}_{3}}\to {{C}_{3}}-{{C}_{1}}\]} Þ \[a(b-1)(c-1)-(b-1)(1-a)-(c-1)(1-a)=0\] Þ \[\frac{a}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=0\] {Divide by\[(1-a)(1-b)(1-c)\]} Þ \[\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=1\].
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