Question

If the lines $ 3x+4y+1=0 $ , $ 5x+\lambda y+3=0 $ and $ 2x+y-1=0 $ are concurrent, then $ \lambda $ is equal to

• A. $ -8 $
• B. $ 8 $
• C. $ 4 $
• D. $ -4 $

Answer 1

Answer: (B)

If three lines are concurrent, then the intersecting point of two lines lies on the third line. Given equation of lines are
$ 3x+4y+1=0 $ ... (i) $ 5x+\lambda y+3=0 $ ... (ii)
and $ 2x+y-1=0 $ ... (iii)
The intersecting point of lines (i) and (iii) is
$ (1,\,\,-1) $ .
Since, the lines are concurrent, therefore the intersecting point
$ (1,\,\,-1) $ lies on line (ii).
$ \therefore $ $ 5(1)+\lambda (-1)+3=0 $
$ \Rightarrow $ $ \lambda =8 $
Alternative Since, the given lines are concurrent.
$ \therefore $ $ \left| \begin{matrix} 3 & 4 & 1 \\ 5 & \lambda & 3 \\ 2 & 1 & -1 \\ \end{matrix} \right|=0 $
$ \Rightarrow $ $ 3(-\lambda -3)-4(-5-6)+1(5-2\lambda )=0 $
$ \Rightarrow $ $ -3\lambda -9+20+24+5-2\lambda =0 $
$ \Rightarrow $ $ -5\lambda +40=0 $
$ \Rightarrow $ $ \lambda =8 $