If the origin of a coordinate system is shift | vedclass

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Question

(C) Let the new coordinates of the point be $(X, Y)$.Given that the origin is shifted to $(h, k) = (-\sqrt{2}, \sqrt{2})$ and the axes are rotated ant

Vedclass · Accepted Answer

$(0, -2-\sqrt{2})$

Vedclass · Answer

(C) Let the new coordinates of the point be $(X, Y)$.Given that the origin is shifted to $(h, k) = (-\sqrt{2}, \sqrt{2})$ and the axes are rotated anti-clockwise by an angle $	heta = 45^{\circ} = \frac{\pi}{4}$.The original coordinates are $(x, y) = (1, -1)$.The transformation formulas for new coordinates $(X, Y)$ are:$X = (x - h) \cos 	heta + (y - k) \sin 	heta$$Y = -(x - h) \sin 	heta + (y - k) \cos 	heta$Substituting the values:$X = (1 - (-\sqrt{2})) \cos(45^{\circ}) + (-1 - \sqrt{2}) \sin(45^{\circ})$$X = (1 + \sqrt{2}) \frac{1}{\sqrt{2}} + (-1 - \sqrt{2}) \frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} + 1 - \frac{1}{\sqrt{2}} - 1 = 0$$Y = -(1 + \sqrt{2}) \sin(45^{\circ}) + (-1 - \sqrt{2}) \cos(45^{\circ})$$Y = -(1 + \sqrt{2}) \frac{1}{\sqrt{2}} + (-1 - \sqrt{2}) \frac{1}{\sqrt{2}}$$Y = -\frac{1}{\sqrt{2}} - 1 - \frac{1}{\sqrt{2}} - 1 = -2 - \frac{2}{\sqrt{2}} = -2 - \sqrt{2}$Thus,the new coordinates are $(0, -2-\sqrt{2})$.

If the origin of a coordinate system is shifted to $(-\sqrt{2}, \sqrt{2})$ and the coordinate system is rotated anti-clockwise through an angle $45^{\circ}$,then the point $P(1, -1)$ in the original system has new coordinates

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