The point (4,1) undergoes the following trans | vedclass

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(C) Step $1$: Reflection of $(4,1)$ about the line $y=x$ gives $(1,4)$.Step $2$: Translation of $(1,4)$ by $2$ units in the positive $X$-direction giv

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$(-\sqrt{2}, 7\sqrt{2})$

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(C) Step $1$: Reflection of $(4,1)$ about the line $y=x$ gives $(1,4)$.Step $2$: Translation of $(1,4)$ by $2$ units in the positive $X$-direction gives $(1+2, 4) = (3,4)$.Step $3$: Rotation of $(x,y) = (3,4)$ by an angle $	heta = \frac{\pi}{4}$ anticlockwise about the origin is given by the transformation:$x' = x \cos 	heta - y \sin 	heta$$y' = x \sin 	heta + y \cos 	heta$Substituting the values:$x' = 3 \cos \frac{\pi}{4} - 4 \sin \frac{\pi}{4} = 3 \left(\frac{1}{\sqrt{2}}ight) - 4 \left(\frac{1}{\sqrt{2}}ight) = -\frac{1}{\sqrt{2}}$$y' = 3 \sin \frac{\pi}{4} + 4 \cos \frac{\pi}{4} = 3 \left(\frac{1}{\sqrt{2}}ight) + 4 \left(\frac{1}{\sqrt{2}}ight) = \frac{7}{\sqrt{2}}$Thus,the final position is $\left(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}ight)$.

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