The origin is translated to $(1,2)$. The point $(7,5)$ in the old system undergoes the following…
The origin is translated to $(1,2)$. The point $(7,5)$ in the old system undergoes the following transformations successively.
I. Moves to the new point under the given translation of origin.
II. Translated through 2 units along the negative direction of the new $X$-axis.
III. Rotated through an angle $\frac{\pi}{4}$ about the origin of new system in the clockwise direction. The final position of the point $(7,5)$ is
Under the translation of origin to $(1,2)$ the point $(7,5)$ undergoes to $(7-1,5-2) \equiv(6,3)$
Under the translation through 2 units along the negative direction of the new $x$-axis, the point $(6,3)$ undergoes to $(6-2,3) \equiv(4,3)$
Under the rotation throw an angle $\frac{\pi}{4}$ about the origin of new system in the clockwise direction, the final position of point $(7,5)$
$
\begin{aligned}
& =\left(4 \cos \frac{\pi}{4}+3 \sin \frac{\pi}{4},-4 \sin \frac{\pi}{4}+3 \cos \frac{\pi}{4}\right) \\
& =\left(\frac{4}{\sqrt{2}}+\frac{3}{\sqrt{2}},-\frac{4}{\sqrt{2}}+\frac{3}{\sqrt{2}}\right)=\left(\frac{7}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right)
\end{aligned}
$