When the origin is shifted to (1,–2) by translation of coordinate axes, the transformed coordinates of (3,–2) are ($\alpha ,\beta$). If the axes are rotated about origin through an angle of ${45}^0$ after the translation, then the
transformed coordinates of ($\alpha ,\beta$) are

$$\begin{gathered} \mathrm{Given} (\mathrm{h}, \mathrm{k})=(1, -2), (\mathrm{x}, \mathrm{y})=(3, -2) \mathrm{and} (X, Y)=(\mathrm{\alpha }, \mathrm{\beta }) \\ \mathrm{Now} \mathrm{X}=\mathrm{x}-\mathrm{h}, \mathrm{Y}=\mathrm{y}-\mathrm{k} \\ \Rightarrow \mathrm{\alpha }=3-1, \mathrm{\beta }=-2+2 \\ \Rightarrow \mathrm{\alpha }=2, \mathrm{\beta }=0 \\ \mathrm{So}, (\mathrm{\alpha }, \mathrm{\beta })=(2, 0) \end{gathered}$$

Now axes are rotated through an angle 45⁰ then transformed coordinates of $(\mathrm{\alpha }, \mathrm{\beta }) \mathrm{are} (X, Y)$

$$\begin{gathered} \mathrm{Now} \mathrm{X}=2\cos {45}^0+0(\sin {45}^0) \\ =2\left(\frac{1}{\sqrt{2}}\right)+0 \\ \mathrm{X}=\sqrt{2} \\ \mathrm{and} \mathrm{Y}=-2\sin {45}^0+0\left(\cos {45}^{\circ }\right) \\ =-2\left(\frac{1}{\sqrt{2}}\right)=-\sqrt{2} \\ \therefore (X, Y)=\left(\sqrt{2}, -\sqrt{2}\right) \end{gathered}$$