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

Answer: (A)

$A=G$

Answer 2

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 ${{G}^{2}}={{a}^{2}}$
$\Rightarrow$ ${{G}^{2}}={{A}^{2}}\Rightarrow G=A$