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.
$\therefore$ $\left|\begin{array}{ccc}3& 4& 1\\ 5& \lambda & 3\\ 2& 1& -1\end{array}\right|=0$
$\Rightarrow$ $3(-\lambda -3)-4(-5-6)+1(5-2\lambda )=0$
$\Rightarrow$ $-3\lambda -9+20+24+5-2\lambda =0$
$\Rightarrow$ $-5\lambda +40=0$
$\Rightarrow$ $\lambda =8$