Question

If the lines $x+2ay+a=0,x+3by+b=0$ and $x+4cy+c=0$ 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 $\left|\begin{array}{ccc}1& 2a& a\\ 1& 3b& b\\ 1& 4c& c\end{array}\right|=0$
$\Rightarrow$ $\left|\begin{array}{ccc}1& 2a& a\\ 0& 3b-2a& b-a\\ 0& 4c-2a& c-a\end{array}\right|=0$
$\Rightarrow$ $1[3b-2a)(c-a)-(b-a)(4c-2a)]=0$
$\Rightarrow$ $3bc-3ab-2ac+2a^2-4bc+2ab$
$+4ac-2n^2=0$
$\Rightarrow$ $-bc-ab+2ac=0$
$\Rightarrow$ $bc+ab=2ac$
$\Rightarrow$ $\frac{1}{a}+\frac{1}{c}=\frac{2}{b}$
$\Rightarrow$ $\frac{1}{a},\frac{1}{b},\frac{1}{c}$ are in AP.