Determinants — MATHS STD 12 Science — Question

$\therefore$ Area of triangle = $\frac{1}{2}\left| {\begin{array}{*{20}{c}} {{x_1}}&{{y_1}}&1 \\ {{x_2}}&{{y_2}}&1 \\ {{x_3}}&{{y_3}}&1 \end{array}} \right| = 0$

$\Rightarrow$  $\frac{1}{2}\left| {\begin{array}{*{20}{c}} x&y&1 \\ 1&2&1 \\ 3&6&1 \end{array}} \right| = 0$

$\Rightarrow \frac{1}{2}\left[ {x\left( {2 - 6} \right) - y\left( {1 - 3} \right) + 1\left( {6 - 6} \right)} \right] = 0$

$\Rightarrow - 4x + 2y = 0$

$\Rightarrow - 2x + y = 0$

$ \Rightarrow y = 2x$ which is required line.