Question

The A.M between $m$ and $n$ and the G.M between $a$ and $b$ are each $=\frac{ma+nb}{m+n}$. Find $m$ and $n$ in terms of $a$ and $b$.

• A. $m=\frac{2a\sqrt{b}}{\sqrt{a}+\sqrt{b}}$, $n=\frac{2b\sqrt{a}}{\sqrt{a}+\sqrt{b}}$
• B. $m=\frac{2a\sqrt{a}}{\sqrt{a}+\sqrt{b}}$, $n=\frac{2b\sqrt{b}}{\sqrt{a}+\sqrt{b}}$
• C. $m=\frac{2b\sqrt{b}}{\sqrt{a}+\sqrt{b}}$, $n=\frac{2a\sqrt{a}}{\sqrt{a}+\sqrt{b}}$
• D. $m=\frac{2b\sqrt{a}}{\sqrt{a}+\sqrt{b}}$, $n=\frac{2a\sqrt{b}}{\sqrt{a}+\sqrt{b}}$

Answer 1

The correct option is D $m=\frac{2b\sqrt{a}}{\sqrt{a}+\sqrt{b}}$, $n=\frac{2a\sqrt{b}}{\sqrt{a}+\sqrt{b}}$

$A.M$ of $m$ and $n$ is $\frac{m+n}{2}=\sqrt{ab}=\frac{ma+nb}{m+n}$

$\Rightarrow m+n=2\sqrt{ab}$ ...(1)

and $ma+nb=(m+n)\sqrt{ab}\sqrt{ab}=2ab$ ...(2)

From equation (1)

$ma+na=2a\sqrt{ab}$ ...(3)

Equation (2)-(3) gives:

$n(b-a)=2ab-2a\sqrt{ab}$

$\Rightarrow n(b-a)=2a\sqrt{b}(\sqrt{b}-\sqrt{a})$

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