A) \[A=G\]
B) \[A=2G\]
C) \[2A=G\]
D) \[{{A}^{2}}=G\]
E) \[A={{G}^{2}}\]
Correct Answer: A
Solution :
Let\[\alpha \]and\[\beta \]are the roots of the equation \[{{x}^{2}}-2ax+{{a}^{2}}=0.\] \[\therefore \]\[\alpha +\beta =2a\]and \[\alpha \beta ={{a}^{2}}\] ...(i) Since, A and G are the arithmetic and geometric mean of the roots. ie,\[A=\frac{\alpha +\beta }{2}\]and \[G=\sqrt{\alpha \beta }\] \[\therefore \]From Eq. (i), \[\frac{\alpha +\beta }{2}=a\]and\[\alpha \beta ={{a}^{2}}\] \[\Rightarrow \]\[A=a\] and\[{{G}^{2}}={{a}^{2}}\] \[\Rightarrow \] \[{{G}^{2}}={{A}^{2}}\Rightarrow G=A\]
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