Question

If the $A.M$. of two numbers a and $b,a>b>0$ is twice their $G.M..$ then $\frac{a}{b}=$

• A. $2+\sqrt{3}$
• B. $7+2\sqrt{3}$
• C. $4+2\sqrt{3}$
• D. $7+4\sqrt{3}$

Answer 1

Answer: (D)

$7+4\sqrt{3}$

Let the larger number be $x$ and smaller be $y$

Given, $A.M.=2G.M$

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

Squaring both sides,we get

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

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

$\Rightarrow a^2+2ab+b^2=16ab$

$\Rightarrow a^2+b^2=16ab-2ab=14ab$

$\Rightarrow a^2+b^2-14ab=0$

Dividing by $b^2$ throughout we get

$\Rightarrow \frac{a^2}{b^2}+1-14\frac{a}{b}=0$

$\Rightarrow \frac{a^2}{b^2}-14\frac{a}{b}+1=0$

$\Rightarrow \frac{a}{b}=\frac{14\pm \sqrt{196-4\times 1\times 1}}{2}$

$=\frac{14\pm \sqrt{192}}{2}=\frac{14\pm 8\sqrt{3}}{2}=7+4\sqrt{3}$