Question

If the arithmetic mean of two numbers a and $b,a>b>0$, is five times their geometric mean, then $\frac{a+b}{a-b}$ is equal to.

• A. $\frac{7\sqrt{3}}{12}$
• B. $\frac{2\sqrt{2}}{4}$
• C. $\frac{\sqrt{6}}{2}$
• D. $\frac{5\sqrt{6}}{12}$

Answer 1

The correct option is D $\frac{5\sqrt{6}}{12}$

$\Rightarrow$$a$ and $b$ are the two positive numbers.

$\Rightarrow$Arithmetic Mean $=\frac{a+b}{2}$

$\Rightarrow$Geometric Mean $=\sqrt{ab}$

$\Rightarrow \frac{a+b}{2}=5\sqrt{ab}$

$\Rightarrow a+b=10\sqrt{ab}$

$\Rightarrow (a+b{)}^2=100ab$

$\Rightarrow$$(a-b{)}^2=(a+b{)}^2-4ab=96ab$

Now,

$\Rightarrow$$\frac{a+b}{a-b}=\sqrt{\left(\frac{(a+b{)}^2}{(a-b{)}^2}\right)}=\sqrt{\frac{100}{96}}$

$=\frac{5\sqrt{6}}{12}$