| If \[x=a\,\,\sin \theta -b\,\,\cos \theta ,\]\[y=a\,\,\cos \theta +b\,\,\sin \theta ,\] then which of the following is true? |
A) \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}+{{b}^{2}}\]
B) \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\]
C) \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}-{{b}^{2}}\]
D) \[\frac{{{x}^{2}}}{{{y}^{2}}}+\frac{{{a}^{2}}}{{{b}^{2}}}=1\]
Correct Answer: A
Solution :
| We have, \[x=a\sin \theta -b\cos \theta \] |
| \[\therefore \] \[{{x}^{2}}={{a}^{2}}{{\sin }^{2}}\theta +{{b}^{2}}\cos \theta -2ab\sin \theta \cdot \cos \theta \] |
| Similarly, \[{{y}^{2}}={{a}^{2}}{{\cos }^{2}}\theta +{{b}^{2}}{{\sin }^{2}}\theta +2ab\sin \theta \cdot \cos \theta \] |
| \[\therefore \] \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}+{{b}^{2}}\] |
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