Question

$a$ and $b$ $(a>b)$ are two positive numbers. Their H.M and G.M are in the ratio $4:5$. Then $a:b$ is

• A. $1:4$
• B. $4:1$
• C. $4:5$
• D. $2:1$

Answer 1

The correct option is B $4:1$

G.M of $a$ and $b$ =$\sqrt{ab}$

H.M of $a$ and $b=\frac{2ab}{a+b}$

Taking the ratio of both H.M and G.M, we get

$\frac{2ab}{(a+b)\left(\sqrt{ab}\right)}=\frac{2\sqrt{ab}}{a+b}=\frac{4}{5}$

Squaring both sides, we get

$\frac{4ab}{(a+b{)}^2}=\frac{16}{25}$

$25ab=4a^2+8ab+4b^2$

$4a^2-17ab+4b^2=0$

$4\frac{a^2}{b^2}-17\frac{a}{b}+4=0$

On solving, we get

$\frac{a}{b}=\frac{1}{4}$ and $4$

Since, $a>b$ therefore $\frac{a}{b}=4$

Hence, option 'B' is correct.