If the $A . M$ is twice the $G . M .$ of the numbers $a$ and $b$ , then $a : b$ will be

\frac{2 - \sqrt{3}}{2 + \sqrt{3}}

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\frac{2 + \sqrt{3}}{2 - \sqrt{3}}

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\frac{\sqrt{3} - 2}{\sqrt{3} + 2}

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\frac{\sqrt{3} + 2}{\sqrt{3} - 2}

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Solution

The correct options are
A $\frac{2 - \sqrt{3}}{2 + \sqrt{3}}$
B $\frac{2 + \sqrt{3}}{2 - \sqrt{3}}$
If a and b are in gp then let a =ar and b=a/r

$A M = 2 G M$

$a + b / 2 = 2 \sqrt a b$

$a + b = 4 \sqrt a b$

Put $a$ and $b$ in this

$a r + a / r = 4 ✓ (a r) (a / r)$

$a (r + 1 / r) = 4 \sqrt a^{2}$

$r + \frac{1}{r} = 4$

$r^{2} - 4 r + 1 = 0$

$r = 2 + \sqrt 3, 2 - \sqrt 3$
$a = 2 + \sqrt 3, b = 2 - \sqrt 3$
Now we have

$a : b = \frac{2 + ✓ 3}{2 - ✓ 3}$

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