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(D) Let $(x, y)$ be the coordinates in the original system and $(x', y')$ be the coordinates in the new system after rotation by an angle $	heta = 13

Vedclass · Accepted Answer

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

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(D) Let $(x, y)$ be the coordinates in the original system and $(x', y')$ be the coordinates in the new system after rotation by an angle $	heta = 135^{\circ}$.The transformation equations are:$x' = x \cos 	heta + y \sin 	heta$$y' = -x \sin 	heta + y \cos 	heta$Given $(x', y') = (4, -3)$ and $	heta = 135^{\circ}$,we have:$4 = x \cos 135^{\circ} + y \sin 135^{\circ} = -\frac{x}{\sqrt{2}} + \frac{y}{\sqrt{2}}$$\Rightarrow -x + y = 4\sqrt{2} \quad (i)$$-3 = -x \sin 135^{\circ} + y \cos 135^{\circ} = -\frac{x}{\sqrt{2}} - \frac{y}{\sqrt{2}}$$\Rightarrow x + y = 3\sqrt{2} \quad (ii)$Adding $(i)$ and $(ii)$ gives $2y = 7\sqrt{2} \Rightarrow y = \frac{7}{\sqrt{2}}$.Subtracting $(i)$ from $(ii)$ gives $2x = -\sqrt{2} \Rightarrow x = -\frac{1}{\sqrt{2}}$.Thus,the coordinates in the original system are $(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}})$.

The coordinate axes are rotated through an angle $135^{\circ}$. If the coordinates of a point $P$ in the new system are known to be $(4, -3)$,then the coordinates of $P$ in the original system are

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