Question

If $A$ be the $A.M.$ and $H$ be the $H.M.$ of two numbers $a$ and $b,$ then the value of $\left(\frac{a-A}{a-H}\right)\cdot \left(\frac{b-A}{b-H}\right)$ is

• A. $\frac{H}{A}$
• B. $H$
• C. $\frac{A}{H}$
• D. $A$

Answer 1

The correct option is C $\frac{A}{H}$

Given,
$A=\frac{a+b}{2}\cdots \left(i\right)$
and
$H=\frac{2ab}{a+b}\cdots \left(ii\right)$
Now,
$\frac{a-A}{a-H}.\frac{b-A}{b-H}=\frac{a-\frac{a+b}{2}}{a-\frac{2ab}{a+b}}.\frac{b-\frac{a+b}{2}}{b-\frac{2ab}{a+b}}=\frac{(a-b)(b-a)(a+b{)}^2}{4(a^2-ab)(b^2-ab)}=\frac{(a+b{)}^2}{4ab}=\left(\frac{a+b}{2}\right)\left(\frac{a+b}{2ab}\right)=\frac{A}{H}$