Question

If the lines $ x+2ay+a=0,\,\,\,\,x+3by+b=0 $ and $ x+4\,cy+c=0 $ are concurrent, where a, b, c are non-zero real numbers, then

• A. $ \frac{1}{a},\frac{1}{b},\frac{1}{c} $ are in an AP
• B. $ \frac{1}{a},\frac{1}{b},\frac{1}{c} $ are in a GP
• C. a, b, c are in an AP
• D. a, b, c are in a GP

Answer 1

Answer: (A)

Given lines are concurrent, if $ \left| \begin{matrix} 1 & 2a & a \\ 1 & 3b & b \\ 1 & 4c & c \\ \end{matrix} \right|=0 $
$ \Rightarrow $ $ \left| \begin{matrix} 1 & 2a & a \\ 0 & 3b-2a & b-a \\ 0 & 4c-2a & c-a \\ \end{matrix} \right|=0 $
$ \Rightarrow $ $ 1[3b-2a)(c-a)-(b-a)(4c-2a)]=0 $
$ \Rightarrow $ $ 3bc-3ab-2ac+2{{a}^{2}}-4bc+2ab $
$ +4ac-2{{n}^{2}}=0 $
$ \Rightarrow $ $ -bc-ab+2ac=0 $
$ \Rightarrow $ $ bc+ab=2ac $
$ \Rightarrow $ $ \frac{1}{a}+\frac{1}{c}=\frac{2}{b} $
$ \Rightarrow $ $ \frac{1}{a},\frac{1}{b},\frac{1}{c} $ are in AP.