A) \[{{x}^{2}}+{{y}^{2}}-{{z}^{2}}\]
B) \[{{x}^{2}}-{{y}^{2}}-{{z}^{2}}\]
C) \[{{x}^{2}}-{{y}^{2}}+{{z}^{2}}\]
D) \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}\]
Correct Answer: A
Solution :
Here, \[z=x\cos \theta +y\sin \theta \] \[{{z}^{2}}={{x}^{2}}{{\cos }^{2}}\theta +{{y}^{2}}{{\sin }^{2}}\theta +2xy\sin \theta \cos \theta \] \[\Rightarrow \,\,2xy\,\sin \theta \cos \theta ={{z}^{2}}-{{x}^{2}}{{\cos }^{2}}\theta -{{y}^{2}}{{\sin }^{2}}\theta \] Let, \[L={{(x\sin \theta -y\cos \theta )}^{2}}\] \[\Rightarrow \,\,L={{x}^{2}}{{\sin }^{2}}\theta +{{y}^{2}}{{\cos }^{2}}\theta -[{{z}^{2}}-{{x}^{2}}{{\cos }^{2}}\theta -{{y}^{2}}{{\sin }^{2}}\theta ]\]\[\Rightarrow \,\,L={{x}^{2}}[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta ]+{{y}^{2}}[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta ]-{{z}^{2}}\]\[\Rightarrow L={{x}^{2}}+{{y}^{2}}-{{z}^{2}}\]
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