Relation between A.M., G.M. and H.M. - Practice Questions & MCQ

Properties of A.M. and G.M
A and G are arithmetic and geometric mean of ' $a$ ' and ' $b$ ', two real, positive and distinct numbers. Then,
$a$ and $b$ are the roots of the equation $x^2-2 A x+G^2=0$.
a and b are given by $A \pm \sqrt{(A+G)(A-G)}$.

Proof:

$
\begin{aligned}
& A=\frac{a+b}{2} \Rightarrow 2 A=a+b \\
& G=\sqrt{a b} \Rightarrow G^2=a b
\end{aligned}
$

$a$ and $b$ are the roots of the equation, then

$
\begin{aligned}
& x^2-2(\text { sum of roots }) x+\text { products of roots }=0 \\
& \Rightarrow x^2-(a+b)+a b=0 \\
& \Rightarrow x^2-2 A x+G^2=0
\end{aligned}
$
The roots of the equation are

$
\begin{aligned}
& x=\frac{2 A \pm \sqrt{(-2 A)^2-4 \cdot 1 \cdot G^2}}{2} \\
& x=A \pm \sqrt{(A+G)(A-G)}
\end{aligned}
$