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
Correct Answer: A
Solution :
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.
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