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

#### Options

0

1

2

Infinite

#### Solution

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.

\[\therefore \begin{vmatrix}1 & - 2 & 3 \\ \lambda & 3 & 1 \\ 4 & - \lambda & 2\end{vmatrix} = 0\]

\[ \Rightarrow \left( 6 + \lambda \right) + 2\left( 2\lambda - 4 \right) + 3\left( - \lambda^2 - 12 \right) = 0\]

\[ \Rightarrow 6 + \lambda + 4\lambda - 8 - 3 \lambda^2 - 36 = 0\]

\[ \Rightarrow 5\lambda - 3 \lambda^2 - 38 = 0\]

\[ \Rightarrow 3 \lambda^2 - 5\lambda + 38 = 0\]

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.