A) \[\sqrt{5}{{X}^{2}}-4XY+{{Y}^{2}}={{r}^{2}}\]
B) \[{{X}^{2}}+2XY-\sqrt{5}{{Y}^{2}}={{r}^{2}}\]
C) \[{{X}^{2}}-{{Y}^{2}}={{r}^{2}}\]
D) \[{{X}^{2}}+{{Y}^{2}}={{r}^{2}}\]
Correct Answer: D
Solution :
Given equation is\[{{x}^{2}}+{{y}^{2}}={{r}^{2}}\]. After rotation \[x=X\cos {{36}^{o}}-Y\sin {{36}^{o}}\] and \[y=X\sin {{36}^{o}}+Y\cos {{36}^{o}}\] \[\therefore \]\[{{X}^{2}}({{\cos }^{2}}{{36}^{o}}+{{\sin }^{2}}{{36}^{o}})\] \[+{{Y}^{2}}({{\sin }^{2}}{{36}^{o}}+{{\cos }^{2}}{{36}^{o}})={{r}^{2}}\] \[\Rightarrow \] \[{{X}^{2}}+{{Y}^{2}}={{r}^{2}}\]
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