Question

The A.M. of two numbers is 3 times their G.M. Show that the numbers are in the ratio

$3 plus 2 square root of 2 colon space space 3 minus 2 square root of 2.$

Solution

Let a and b be two numbers.
 A.M = $fraction numerator straight a plus straight b over denominator 2 end fraction comma$ G.M. = $space space square root of ab$
According to the given condition $fraction numerator straight a plus straight b over denominator 2 end fraction space equals space 3 square root of ab space rightwards double arrow space fraction numerator straight a plus straight b over denominator 2 square root of ab end fraction space equals space 3 over 1$
Applying componendo and dividendo, we get
 $fraction numerator straight a plus straight b plus 2 square root of ab over denominator straight a plus straight b minus 2 square root of ab end fraction space equals space fraction numerator 3 plus 1 over denominator 3 minus 1 end fraction space space space rightwards double arrow space space space space space fraction numerator left parenthesis square root of straight a plus square root of straight b right parenthesis squared over denominator left parenthesis square root of straight a minus square root of straight b right parenthesis end root squared end fraction space equals space 4 over 2$
$rightwards double arrow$ $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
Again applying componendo and dividendo, we get
$fraction numerator square root of straight a plus square root of straight b plus square root of straight a minus square root of straight b over denominator square root of straight a plus square root of straight b minus square root of straight a plus square root of straight b end fraction space equals space fraction numerator 2 plus square root of 2 over denominator 2 minus square root of 2 end fraction space space space space rightwards double arrow space space space space space fraction numerator 2 square root of straight a over denominator 2 square root of straight b end fraction space equals space fraction numerator 2 plus square root of 2 over denominator 2 minus square root of 2 end fraction$

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Squaring both sides, we get
 $straight a over straight b space equals space fraction numerator 4 plus 2 plus 2 left parenthesis 2 right parenthesis square root of 2 over denominator 4 plus 2 minus 2 left parenthesis 2 right parenthesis square root of 2 end fraction$
$rightwards double arrow$ $straight a over straight b space equals space fraction numerator 6 plus 4 square root of 2 over denominator 6 minus 4 square root of 2 end fraction space equals space fraction numerator 3 plus 2 square root of 2 over denominator 3 minus 2 square root of 2 end fraction$