Question

If the $A.M.,G.M\mathrm{and}H.M$ between two positive numbers $a\mathrm{and}b$ are equal, then

• A. $a=b$
• B. $ab=1$
• C. $a>b$
• D. $a<b$

Answer 1

The correct option is A

$a=b$

Explanation for Correct Option:

Step 1: Apply the formula

$AM=\frac{(a+b)}{2}$

$GM=\sqrt{\left(ab\right)}$

$HM=\frac{2ab}{(a+b)}$

Let $a$ and $b$ be the $2$ positive numbers.

Since $AM=GM$

$\therefore \frac{(a+b)}{2}=\sqrt{\left(ab\right)}$

Since $AM=HM$

$$\begin{gathered} \therefore \frac{(a+b)}{2}=\frac{2ab}{a+b} \\ \Rightarrow {(a+b)}^2=4ab \end{gathered}$$

Step 2:Since HM = GM we get

$\Rightarrow \frac{2ab}{a+b}=\sqrt{ab}$

On Squaring the above equation

$\Rightarrow {(a+b)}^2=4ab$

$\Rightarrow a^2+2ab+b^2=4ab$

$\Rightarrow a^2-2ab+b^2=0$

${\Rightarrow (a-b)}^2=0$

$\therefore a=b$

Hence option (1) is the answer.