Question

The mean proportional between (2 +$\sqrt{3\\}$)and (2 −$\sqrt{3}$)is:

Answer 1

$\red{ \huge{ \underline{ \overline{ \mid{ \bold{ \purple{ \: \: SOLUTION \: \: \red{ \mid}}}}}}}}$

$\bold{Let \: ,} \\ \\ \bold{ \: \: \: Mean \: \: proportional \: \: be \: = \: x} \\ \\ \bold{ \therefore \: (2 + \sqrt{3}) \: : \: x \: = \: x \: : \: (2 - \sqrt{3} )} \\ \\ \bold{ \Longrightarrow \: \frac{2 + \sqrt{3} }{x} = \frac{x}{2 - \sqrt{3} } } \\ \\ \bold{ \Longrightarrow \: x {}^{2} = (2 + \sqrt{3} )(2 - \sqrt{3} )} \\ \\ \bold{ \Longrightarrow \: = x {}^{2} = (2) {}^{2} - (\sqrt{3} ) {}^{2} } \\ \\ \bold{ \Longrightarrow \: x {}^{2} = 4 - 3} \\ \\ \bold{ \Longrightarrow \: x {}^{2} = 1 } \\ \\ \bold{ \Longrightarrow \: x = \sqrt{1} } \\ \\ \bold{ \therefore \: \: x \: = \: 1}$
$\bold{Hence,} \\ \\ \bold{The \: \: mean \: \: proportional \: \: between \: \: (2 + \sqrt{3} )} \\ \bold{and \: \: (2 - \sqrt{3} ) \: \: is \: \: \red{ \: \: 1 \: \: }.}$

Answer 2

$\huge\mathtt\red{Answer}$

$\begin{lgathered}\bold{Let \: ,} \\ \\ \bold{ \: \: \: Mean \: \: proportional \: \: be \: = \: x} \\ \\ \bold{ \therefore \: (2 + \sqrt{3}) \: : \: x \: = \: x \: : \: (2 - \sqrt{3} )} \\ \\ \bold{ \Longrightarrow \: \frac{2 + \sqrt{3} }{x} = \frac{x}{2 - \sqrt{3} } } \\ \\ \bold{ \Longrightarrow \: x {}^{2} = (2 + \sqrt{3} )(2 - \sqrt{3} )} \\ \\ \bold{ \Longrightarrow \: = x {}^{2} = (2) {}^{2} - (\sqrt{3} ) {}^{2} } \\ \\ \bold{ \Longrightarrow \: x {}^{2} = 4 - 3} \\ \\ \bold{ \Longrightarrow \: x {}^{2} = 1 } \\ \\ \bold{ \Longrightarrow \: x = \sqrt{1} } \\ \\ \bold{ \therefore \: \: x \: = \: 1}\end{lgathered}$

$\begin{lgathered}\bold{Hence,} \\ \\ \bold{The \: \: mean \: \: proportional \: \: between \: \: (2 + \sqrt{3} )} \\ \bold{and \: \: (2 - \sqrt{3} ) \: \: is \: \: \red{ \: \: 1 \: \: }.}\end{lgathered}$