Question

The value of λ for which the lines 3x + 4y = 5, 5x + 4y = 4 and λx + 4y = 6 meet at a point is
(a) 2
(b) 1
(c) 4
(d) 3
(e) 0

Answer 1

(b) 1

It is given that the lines 3x + 4y = 5, 5x + 4y = 4 and λx + 4y = 6 meet at a point.
In other words, the given lines are concurrent.

$$\begin{gathered} \left|\begin{array}{ccc}3& 4& -5\\ 5& 4& -4\\ \lambda & 4& -6\end{array}\right|=0 \\ \Rightarrow 3\left(-24+16\right)-4\left(-30+4\lambda \right)-5\left(20-4\lambda \right)=0 \\ \Rightarrow -24+120-16\lambda -100+20\lambda =0 \\ \Rightarrow 4\lambda =4 \\ \Rightarrow \lambda =1 \end{gathered}$$