If lines 2x - 3y + 6 = 0 and kx + 2y + 12 =0 cut the co-ordinate axes in concyclic points then the value of $\left|\mathrm{k}\right|$ is

Line 2x - 3y + 6 = 0 meets the axe s at A(-3, 0) and B(0,2).
Line kx + 2y + 12 = 0 meets the axes at D(0,-6) and c(-12/k, 0).

since the points A, B, c and D are concyclic, points A and, C cannot be at the same side of origin (see figure).
Also, points A, B, C and D will be concyclic if
$\begin{array}{l}\left|\mathrm{OA}\right|\times \left|\mathrm{OC}\right|=\left|\mathrm{OB}\right|\times \left|\mathrm{OD}\right|\\ \Rightarrow 3\times |12/\mathrm{k}|=2\times 6\\ \Rightarrow \left|\mathrm{k}\right|=3\end{array}$