A = (1, 2), B = (–1, 4) if the axes are rotated through an angle $\frac{\pi }{12}$
in anticlockwise direction then the slope of the line AB with reference to the new axes is

$$\begin{gathered} \mathrm{A}=(1, 2), B=(-1, 4) \\ \mathrm{slope} \mathrm{of} \overline{\mathrm{AB}}=\frac{4-2}{-1-1} \\ \Rightarrow \mathrm{m}=-1 \\ \Rightarrow \mathrm{tan\theta }=\tan {135}^{\circ } \\ \Rightarrow \mathrm{\theta }={135}^0 \end{gathered}$$

$$\begin{gathered} Slope of line \overline{AB} with reference to old axis={135}^0 \\ If the axes are rotated through an angle {15}^{0 }in anticlockwise direction then \\ inclination of line \overline{AB} w.r.to new axes={120}^0 \\ Slope of line \overline{AB} with reference to new axis=\tan {120}^0=-\sqrt{3} \end{gathered}$$