Login/Sign Up
Home » For what value of λ are the three lines 2x – 5y + 3 = 0, 5x – 9y + λ = 0 and x – 2y + 1 = 0 concurrent?
For what value of λ are the three lines 2x – 5y + 3 = 0, 5x – 9y + λ = 0 and x – 2y + 1 = 0 concurrent?
CBSE | Class 11 | Exercise 23.11 | Maths | RD Sharma | Straight Lines
For what value of λ are the three lines 2x – 5y + 3 = 0, 5x – 9y + λ = 0 and x – 2y + 1 = 0 concurrent?

Given:

    \[2x~-~5y\text{ }+\text{ }3\text{ }=\text{ }0\text{ }\ldots \text{ }\left( 1 \right)\]

    \[5x~-~9y\text{ }+~\lambda ~=\text{ }0\text{ }\ldots \text{ }\left( 2 \right)\]

And ,

    \[x~-~2y\text{ }+\text{ }1\text{ }=\text{ }0\text{ }\ldots \text{ }\left( 3 \right)\]

It is given that the three lines are concurrent.

Now, consider the following determinant:

RD Sharma Solutions for Class 11 Maths Chapter 23 – The Straight Lines - image 61

    \[2\left( -9\text{ }+\text{ }2\lambda \right)\text{ }+\text{ }5\left( 5\text{ }~\lambda \right)\text{ }+\text{ }3\left( -10\text{ }+\text{ }9 \right)\text{ }=\text{ }0\]

    \[-18\text{ }+\text{ }4\lambda ~+\text{ }25\text{ }\text{ }5\lambda ~\text{ }3\text{ }=\text{ }0\]

    \[\lambda ~=\text{ }4\]

∴ The value of λ is 4.

i

More Material