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The co-ordinate axis rotated through an angle \( \Large 135 ^{\circ} \). If the co-ordinates of a points P in the new system are known to be \( \Large \left(4,\ -3\right) \), then the co-ordinates of P in the original systems are:


A) \( \Large \left(\frac{1}{\sqrt{2}},\ \frac{7}{\sqrt{2}}\right) \)

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

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

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

Correct Answer:
D) \( \Large \left( - \frac{1}{\sqrt{2}},\ \frac{7}{\sqrt{2}}\right) \)

Description for Correct answer:

We know, if co-ordinate axes are rotated, then

\( \Large P= \left(x \cos \theta - y \sin \theta ,\ x \sin \theta + y \cos \theta \right) \)

Its rotated at an angle \( \Large 135 ^{\circ} i.e.,\ \theta = 135 ^{\circ} \) and new point be

\( \Large P= 4 \cos \left(90 ^{\circ} + 45 ^{\circ} \right) + 3 \sin \left(90 ^{\circ} + 45 ^{\circ} \right), \) \( \Large 4 \sin \left(90 ^{\circ} + 45 ^{\circ} \right) - 3 \cos \left(90 ^{\circ} + 45 ^{\circ} \right) \)

= \( \Large \left[ 4 \left(\frac{-1}{\sqrt{2}}+3.\frac{1}{\sqrt{2}},\ 4.\frac{1}{\sqrt{2}}+3.\frac{1}{\sqrt{2}}\right) \right] = \left(-\frac{1}{\sqrt{2}},\ \frac{7}{\sqrt{2}}\right) \)


Part of solved Rectangular and Cartesian products questions and answers : >> Elementary Mathematics >> Rectangular and Cartesian products




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