Question

If the A.M between a and b is m times their H.M then a:b is :

• A. $\frac{\sqrt{m}+\sqrt{m+1}}{\sqrt{m}-\sqrt{m+1}}$
• B. $\frac{\sqrt{m}-\sqrt{m-1}}{\sqrt{m}+\sqrt{m+1}}$
• C. $\frac{\sqrt{m}+\sqrt{m-1}}{\sqrt{m}-\sqrt{m-1}}$
• D. $\frac{\sqrt{m}-\sqrt{m+1}}{\sqrt{m}+\sqrt{m+1}}$

Answer 1

The correct option is A $\frac{\sqrt{m}+\sqrt{m+1}}{\sqrt{m}-\sqrt{m+1}}$

$Am=\frac{a+b}{2}$

$Hm=\frac{2ab}{a+b}$

$\frac{Am}{Hm}=\frac{(a+b{)}^2}{4ab}=\frac{m}{n}$

let, $m=(a+b{)}^2,n=4ab\Rightarrow m-n=(a-b{)}^2$

$a+b=\sqrt{m}$

$a-b=\sqrt{m-n}$

$\Rightarrow a=\frac{\sqrt{m}+\sqrt{m-n}}{2}$

$\Rightarrow b=\frac{\sqrt{m}-\sqrt{m-n}}{2}$

$\frac{a}{b}=\frac{\sqrt{m}+\sqrt{m-n}}{\sqrt{m}-\sqrt{m-n}}$