Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# 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 - Mathematics

MCQ

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

• 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.

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 23 The straight lines
Q 8 | Page 133