Question

The number of real values of λ for which the lines x − 2y + 3 = 0, λx + 3y + 1 = 0 and 4x − λy + 2 = 0 are concurrent is
(a) 0
(b) 1
(c) 2
(d) Infinite

Answer 1

(a) 0

The given lines are

x − 2y + 3 = 0 ... (1)

λx + 3y + 1 = 0 ... (2)

4x − λy + 2 = 0 ... (3)

It is given that (1), (2) and (3) are concurrent.

$$\begin{gathered} \therefore \left|\begin{array}{ccc}1& -2& 3\\ \lambda & 3& 1\\ 4& -\lambda & 2\end{array}\right|=0 \\ \Rightarrow \left(6+\lambda \right)+2\left(2\lambda -4\right)+3\left(-{\lambda }^2-12\right)=0 \\ \Rightarrow 6+\lambda +4\lambda -8-3{\lambda }^2-36=0 \\ \Rightarrow 5\lambda -3{\lambda }^2-38=0 \\ \Rightarrow 3{\lambda }^2-5\lambda +38=0 \end{gathered}$$

The discriminant of this equation is $25-4\times 3\times 38=-431$

Hence, there is no real value of $\lambda$ for which the lines x − 2y + 3 = 0, λx + 3y + 1 = 0 and 4x − λy + 2 = 0 are concurrent.