Question

If $a$ is positive and if $A$ and $G$ are the arithmetic mean and the geometric mean of the roots of $ {{x}^{2}}-2ax+{{a}^{2}}=0 $ respectively, then

• A. $ A=G $
• B. $ A=2G $
• C. $ 2A=G $
• D. $ {{A}^{2}}=G $

Answer 1

The Correct Option is A

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. i.e,
$ A=\frac{\alpha +\beta }{2} $ and $ G=\sqrt{\alpha \beta } $
$ \therefore $ From E (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 $