First slide

Equation of line in Straight lines

Question

$If the lines 2 x + y - k = 0, x - y + 1 = 0 and - 2 x - 3 y + 8 = 0 are concurrent then k =$

Moderate

Solution

$The condition that the lines a_{1} x + b_{1} y + c_{1} = 0, a_{2} x + b_{2} y + c_{2} = 0 and a_{3} x + b_{3} y + c_{3} = 0 are to$
$be concurrent is |\begin{array}{lll} a_{1} b_{1} c_{1} \\ a_{2} b_{2} c_{2} \\ a_{3} b_{3} c_{3} \end{array}| = 0$
Hence,
$|\begin{matrix} 2 & 1 & - k \\ 1 & - 1 & 1 \\ - 2 & - 3 & 8 \end{matrix}| = 0$
Expand the determinant
$\begin{array}{l} 2 (- 8 + 3) - 1 (8 + 2) - k (- 3 - 2) = 0 \\ - 10 - 10 + 5 k = 0 \\ k = \frac{20}{5} \\ = 4 \end{array}$
$Therefore, k = 4$

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