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Question
The value of λ for which the lines 3x + 4y = 5, 5x + 4y = 4 and λx + 4y = 6 meet at a point is
Options
2
1
4
3
0
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Solution
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.
\[\begin{vmatrix}3 & 4 & - 5 \\ 5 & 4 & - 4 \\ \lambda & 4 & - 6\end{vmatrix} = 0\]
\[ \Rightarrow 3\left( - 24 + 16 \right) - 4\left( - 30 + 4\lambda \right) - 5\left( 20 - 4\lambda \right) = 0\]
\[ \Rightarrow - 24 + 120 - 16\lambda - 100 + 20\lambda = 0\]
\[ \Rightarrow 4\lambda = 4\]
\[ \Rightarrow \lambda = 1\]
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