If the line $y = mx$ meets the lines $x + 2y - 1 = 0$ and $2x - y + 3 = 0$ at the same point, then $

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If the line $y = mx$ meets the lines $x + 2y - 1 = 0$ and $2x - y + 3 = 0$ at the same point, then $m$ is equal to :

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Sol. [Key Concept: Three concurrent lines: substitute the intersection of two lines into the third to solve for $m$.]$y - mx = 0,x + 2y - 1 = 0$ and $2x - y + 3 = 0$ are concurrent$$\Rightarrow \left| \begin{matrix} m & - 1 & 0 \ 1 & 2 & - 1 \ 2 & - 1 & 3 \end{matrix} ight| = 0 = 5m + 5 \Rightarrow m = - 1$$

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