The origin is translated to (1,2). The point | vedclass

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(C) Step $1$: Under the translation of the origin to $(1,2)$,the point $(7,5)$ in the old system becomes $(7-1, 5-2) = (6,3)$ in the new system.Step $

Vedclass · Accepted Answer

$\left(\frac{7}{\sqrt{2}}, \frac{-1}{\sqrt{2}}\right)$

Vedclass · Answer

(C) Step $1$: Under the translation of the origin to $(1,2)$,the point $(7,5)$ in the old system becomes $(7-1, 5-2) = (6,3)$ in the new system.Step $2$: Translating the point $(6,3)$ by $2$ units along the negative direction of the new $X$-axis results in $(6-2, 3) = (4,3)$.Step $3$: Rotating the point $(4,3)$ through an angle $	heta = \frac{\pi}{4}$ clockwise about the origin of the new system. The clockwise rotation formula is $(x', y') = (x \cos 	heta + y \sin 	heta, -x \sin 	heta + y \cos 	heta)$.Substituting $x=4, y=3, 	heta = \frac{\pi}{4}$:$x' = 4 \cos \frac{\pi}{4} + 3 \sin \frac{\pi}{4} = \frac{4}{\sqrt{2}} + \frac{3}{\sqrt{2}} = \frac{7}{\sqrt{2}}$$y' = -4 \sin \frac{\pi}{4} + 3 \cos \frac{\pi}{4} = -\frac{4}{\sqrt{2}} + \frac{3}{\sqrt{2}} = -\frac{1}{\sqrt{2}}$Thus,the final position is $\left(\frac{7}{\sqrt{2}}, -\frac{1}{\sqrt{2}}ight)$.

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