Line \({L}_{1}\) of slope \(2\) and line \({L}_{2}\) of slope \(\frac{1}{2}\) intersect at the origin O . In the first q…

Line \({L}_{1}\) of slope \(2\) and line \({L}_{2}\) of slope \(\frac{1}{2}\) intersect at the origin O . In the first quadrant, \({\mathrm{P}}_{1},{\mathrm{P}}_{2},\ldots .{\mathrm{P}}_{12}\) are 12 points on line \({L}_{1}\) and \({Q}_{1},{Q}_{2},\ldots ..{Q}_{9}\) are \(9\) points on line \({\mathrm{L}}_{2}\). Then the total number of triangles, that can be formed having vertices at three of the \(22\) points \(\mathrm{O},{\mathrm{P}}_{1},{\mathrm{P}}_{2},\ldots {\mathrm{P}}_{12}\), \({\mathrm{Q}}_{1},{\mathrm{Q}}_{2},\ldots .{\mathrm{Q}}_{9}\), is:

[JEE Main 2025, 3 Apr (Shift 2)]